EDEXCEL Maths ➒-➀ GCSE
What is a recurring decimal? (➃)
1) A decimal which has a pattern of numbers that repeat forever, e.g. ¹/₃ = 0.333... 2) It doesn't have to be a single digit that repeats. For instance, you can have: 0.143143143... 3) The repeating part is generally marked with dots on the top. If there is one digit that repeats, there is only one dot. If there are more digits, then everything from the first repeating dot to the end of the repeating pattern will be marked with dots.
What is an algebraic term? (➂)
1) An algebraic term is a collection of numbers, letters and brackets, all multiplied/divided together. 2) These terms are separated by + and - signs. Every term has either + or - in front of it (a term without a sign in front of it is always positive). 3) E.G: 4xy + 5x² - 2y + 6y² + 4. • "4xy" is the 'xy' term. • "5x²" is the 'x²' term. • "-2y" is the 'y' term. • "6y²" is the 'y²' term. • "4" is the 'number' term.
What are the terms you use when showing mathematical proof? (➅)
1) Any even number can be written as '2n' - i.e. 2 multiplied by something. 2) Any odd number can be written as 2n + 1 - i.e. 2 multiplied by something with 1 added on. 3) Consecutive numbers can be written as n, n + 1, n + 2 etc - you can apply this to e.g. consecutive even numbers too (they'd be written as 2n, 2n + 1, 2n + 2 etc.) 4) The sum, difference and product of integers is always an integer.
What's the method to show inequalities on a graph? (➅)
1) Convert each inequality to an equation by simply putting an '=' in place of an inequality sign. 2) Draw the graph for each equation - if the inequality sign is < or >, draw a dotted line; but if it is ≥ or ≤, draw a solid line. 3) Work out which side of each line you want - put a point (usually the origin) into the equality to see if it's on the correct side of the line. 4) Shade the region this gives you.
How do you turn an improper fraction into a mixed number? (➂)
1) Divide the top number by the bottom and this answer will give the integer of the mixed number. 2) The remainder from the division you do will go on top of the fraction and the denominator stays the same. 3) WORKED EXAMPLE: • Write ³¹/₄ as a mixed number. • ³¹/₄ = 7 remainder 3, so ³¹/₄ = 7¾
What does factorising the quadratic mean and what is the standard format for quadratic equations? (➄)
1) Factorising a quadratic means putting it into two brackets. 2) The standard format for quadratic equations is: ax² + bx + c = 0.
How do you find the nth term of a linear sequence? (➃)
1) Find the common difference - this tells you what to multiply n by. 2) Work out what to add or subtract. 3) Put both bits together.
How can you find the nth term of a quadratic sequence? (➐)
1) Find the difference between each pair of terms. 2) The difference is changing, so workout the difference between the differences. 3) Divide this value by 2 - this gives the coefficient of the n² term. 4) Subtract the n² term from each term in the sequence. This will give you a linear sequence. 5) Find the rule for the nth term of a linear sequence and add this onto the n² term.
What is the 7-step method to rearranging formulas? (➄)
1) Get rid of √ signs by squaring both sides. 2) Get rid of any fractions. 3) Multiply out any brackets. 4) Collect all the subject terms on one side. 5) Reduce it to the form 'Ax = B'. 6) Divide both sides by A to give 'x = '. 7) If you have 'x² = ', √ both sides to get 'x = ± '.
How do you estimate square roots? (➄)
1) Find two square numbers, one either side of the number you're given. 2) Decide which it's closest to and make a sensible estimate of the digit after the decimal point. E.G: Estimate the √87. √9 = 81 and √10 = 100. Estimate: 9.3
What are the six steps to solving equations? (➄)
1) Get rid of any fractions. 2) Multiply out any brackets. 3) Put x-terms & number terms on different sides. 4) Reduce to the form 'Ax = B' (combine like terms). 5) Finally, divide both sides by A to give 'x = '. 6) If you had 'x² = ', √ both sides to get 'x = ± '.
How do you round to decimal places? (➂)
1) Identify the position of the last digit from the number of decimal places (last in the rounded version not the original). 2) Then look at the next digit to the right called the decider. 3) If the decider is 5 or more, round up. If it's 4 or less, round down and then leave the last digit as it is. 4) There must be no more digits after the last digit (not even zeroes).
What are the rules for quadratic inequalities? (➑)
1) If x² < a², then: '-a < x < a'. (One answer.) 2) If x² > a², then: 'x > a' OR 'x < -a'. (Two answers.)
What does a sign change indicate? (➐)
1) If you're trying to solve an equation that equals 0, there's an important thing to remember. If there's a sign change (i.e. from a positive to a negative or vice versa) when you put two numbers into the equation, there's a solution between those numbers.
How do you find the minimum and maximum values if the format is a ÷ b? (➐)
1) Let's say a = 5.3 and b= 4.2, both to 1 d.p. What are the maximum values of a ÷ b? • First find the bounds for a and b: 5.25 ≤ a < 5.35 & 4.15 ≤ b < 4.25. • The bigger the number you divide by, the smaller the answer, so: max(a ÷ b) = max(a) ÷ min(b), and min(a÷b) = min(a) ÷ max(b). So: • max(a÷b) = 5.35 ÷ 4.15 = 1.289 (3 d.p.) • min(a÷b) = 5.25 ÷ 4.25 = 1.235 (3.d.p.)
How do you solve an equation with brackets in it? (➃)
1) Multiply out the brackets before rearranging and then solve it in the same way as before. 2) WORKED EXAMPLE: • Solve 3(3x - 2) = 5x + 10 • 9x - 6 = 5x +10 • 9x = 5x + 16 • 4x = 16 • x = 4
What is the method to solve quadratics by completing the square when 'a' ≠ 1? (➒)
1) Rearrange into the standard format: ax² + bx + c = 0. 2) Take out a factor of 'a' from the x² and x-terms. 3) Write out the initial bracket: (x = b/2)² - just divide the value of b by 2. 4)) Multiply out the brackets and compare to the original to find what you need to add or subtract to complete the square. 5) Add or subtract the adjusting number to make it match the original.
How do you solve an equation with a fraction in it? (➄)
1) To get rid of fractions, multiply every term of the equation by whatever's on the bottom of the fraction. If there are two fractions, you'll need to multiply by both denominators. 2) WORKED EXAMPLE: • Solve x + 2/4 = 4x - 7 • (x+2) = 4(4x) - 4(7) (multiply every term by the denominator (4) to get rid of the fraction. • x + 2 = 16x - 28 • 30 = 15x • x = 2
How can you show inequalities on number lines? (➃)
1) Use an open circle (〇) for > or <; and go left for > and right for <. 2) Use a shaded circle (⬤) for ≥ or ≤ and go left for ≥ and right for ≤. 3) Use an open circle (〇) if there is a ≠ sign involved and go both sides of the number line.
What is the method to solving quadratics by completing the square? (➑)
1) Rearrange into the standard format: ax² + bx + c = 0. 2) Write out the initial bracket: (x = b/2)² - just divide the value of b by 2. 3) Multiply out the brackets and compare to the original to find what you need to add or subtract to complete the square. 4) Add or subtract the adjusting number to make it match the original.
How can you prove that the sum of any three odd numbers is odd? (➅)
1) Take three odd numbers: • 2a + 1, 2b + 1, 2c + 1. (They don't have to be consecutive.) 2) Add them together: • (2a + 1) + (2b + 1) + (2c + 1) • = 2a + 2b + 2c + 2 + 1. • = 2(a + b + c + 1) + 1. • = 2n + 1 where an integer (a + b + c + 1). • So the sum of any three numbers is odd.
How do you tackle typical algebra questions with inequalities that require you to find all the possible values of x? (➄)
1) WORKED EXAMPLE: 'x' is an integer such that -4 < x ≤ 3. Find all the possible values of 'x'. • What each bit of the inequality is telling you? • -4 < x means 'x' is greater than -4. • x ≤ 3 means that 'x' is less than or equal to 3. • Now write down all the values that 'x' can take. • = -3, -2, -1, 0, 1, 2, 3.
How do you tackle a typical Fibonacci sequence question? (➂)
1) WORKED EXAMPLE: Find the next two terms in each in the following sequence. • 1, 1, 2, 3, 5, ... • The rule for this sequence is 'add together the two previous terms', so the next two terms are: 3 + 5 = 8 and 5 + 8 = 13. • The answer is: 8 and 13.
How do you tackle a typical geometric sequence question? (➂)
1) WORKED EXAMPLE: Find the next two terms in each in the following sequence. • 0.2, 0.6, 1.8, 5.4, 16.2, ... • This is an example of a geometric progression - there is a common ratio where you multiply or divide by the same number each time. • Common ratio = 0.6 ÷ 0.2 = 3 • 16.2 x 3 = 48.6 and 48.6 x 3 = 145.8 • The next two terms are: 48.6 and 145.8
How do you tackle typical algebra questions with two inequalities that require you to solve the equation? (➄)
1) WORKED EXAMPLE: Solve: -2 ≤ x/4 + 3 ≤ 5. • For this question, you have to do the same thing to each bit of the inequality. • Subtract 3: -2 (-3) ≤ x/4 + 3 (- 3) ≤ 5 (- 3) • = -5 ≤ x/4 ≤ 2 • Multiply by 4: -5 (x 4) ≤ x/4 (x 4) ≤ 2 (x 4) • = -20 ≤ x ≤ 8
How do you tackle typical algebra questions with inequalities that require you to solve the equation and flip the inequality signs? (➄)
1) WORKED EXAMPLE: Solve: 9 - 2x > 15. • You have to - 9: 9 - 2x (-9) > 15 (- 9) • = - 2x > 6 • You have to divide by -2: 2x (÷ -2) > 6 (÷ -2) • = x < -3 (The > turns into a < because we divided by a negative number.)
How do you decide if a term is in a sequence? (➃)
1) WORKED EXAMPLE: • The nth term of a sequence is given by n² - 2. • a) Find the 6th term in the sequence. • This type of question is easy - just put n=6 into the expression: 6² - 2 = 36 - 2 = 34. • b) Is 45 a term in the sequence? • Set it equal to 45: n² - 2 = 45 • n² = 47. Now just solve for n. • n = √47 = 6.8556... • n is not a whole number, so 45 is NOT in the sequence.
How do you turn a recurring decimal with two or more repeating digits into a fraction? (➐)
1) WORKED EXAMPLE: • Write 0.1666... as a fraction. • Name your decimal (let's call it "x"). • Multiply 'x' by a power of ten to move the non-repeating part past the decimal point. • Now multiply by a power of ten to move one full lump past the decimal point. • Subtract to get rid of the fraction part. • Divide to leave 'x', and cancel if possible.
How can you use the 6-step method to solve typical equations? (➄)
1) WORKED EXAMPLE: • Solve 3x + 4/5 = 4x -1/3 = 14 • 5 x 3 x(3x+4)/5 = 5 x 3 x(4x - 1)/3 = 5 x 3 x(14) (Get rid of fractions). • 3(3x + 4) + 5(4x - 1) = 210 • = 9x + 12 + 20x - 5 = 210 (Get rid of brackets). • 9x + 20x = 210 - 12 + 5 (Collect all x-terms on one side and all number terms on the other). • 29x = 203 (Reduce it to the form 'Ax = B'). • x = 7 (Divide both sides by A to give 'x = ').
How can you use the 6-step method to solve equations with squares? (➄)
1) WORKED EXAMPLE: • There are 75 tiles on a roof. Each row contains three times the number of tiles as each column. How many tiles are there in one column? • (3x) x (x) = 75 • 3x² = 75 • x² = 25 • x = ±5 • In this case, you can ignore the negative square root as you can't have a negative number of tiles on the roof. Therefore, x = 5.
How do you simplify algebraic fractions that contain quadratics? (➅)
1) WORKED EXAMPLE: • Simplify x² - 16/x² + 2x - 8. • Factorise the top using the difference of two squares: x² - 16 = (x + 4)(x - 4). • Factorise the quadratic on the bottom: x² + 2x - 8 = (x - 2)(x + 4). • You now have: (x + 4)(x - 4)/(x - 2)(x + 4). • Cancel out to give: x - 4/x - 2.
How do you solve equations with two fractions? (➄)
1) WORKED EXAMPLE: • Solve 3x + 5/2 = 4x + 10/3 • 2 x 3 x(3x + 5)/2 = 2 x 3(4x + 10)/3 (multiply everything by the two denominators - 2 & 3.) • 3(3x + 5) = 2(4x + 10) • 9x + 15 = 8x + 20 • 9x = 8x + 5 • x = 5
What happens when a measurement is truncated? (➄)
1) When a measurement is truncated to a given unit, the actual measurement can be up to a whole unit bigger, but no smaller. 2) You can truncate a number by chopping off decimal places, so if the mass of the cake was 2.4 truncated to 1 d.p. the interval would be 2.4 kg ≤ x < 2.5 kg.
How do you turn a recurring decimal with one repeating digit into a fraction? (➐)
1) WORKED EXAMPLE: • Write 0.717171... as a fraction. • Name your decimals (lets call it "x"). • Multiply 'x' by a power of ten to move it past the decimal point by one full repeated lump - in this case, that's 100. • Now you can subtract to get rid of the decimal part. • Finally, divide to leave x, and cancel if possible.
What are the two typical quadratic inequality questions? (➑)
1) WORKED EXAMPLES: a) Solve the inequality of x² ≤ 25. • If x² = 25, then x = ±5. • As x² ≤ 25, then: '-5 ≤ x ≤ 5'. b) Solve the inequality x² > 9. • x² = 9, then x = ±3. • As x² > 9, then: 'x < -3' OR 'x > 3'.
How do you find upper and lower bounds? (➄)
1) When a measurement is rounded to a given unit, the actual measurement can be anything up to half a unit bigger or smaller. 2) WORKED EXAMPLE: • The mass of a cake is given as 2.4 kg to the nearest 0.1 kg. Find the interval within which m, the actual mass of the cake lies. • lower bound = 2.4 - 0.05 = 2.35 kg • upper bound = 2.4 + 0.05 = 2.45 kg • Interval = 2.35 kg ≤ m < 2.45 kg
How do you multiply out double brackets? (➃)
Double brackets are trickier than single brackets - this time, you have to multiply out everything in the first bracket by everything in the second bracket. You can use the foil method.
How do you multiply out three brackets? (➐)
For three brackets, you simply multiply two together and then multiply the result by the remaining bracket.
How do you divide fractions? (➂)
Invert the second fraction (turn it into a reciprocal) and multiply. E.G. ¹/₃ ÷ ⅘ = ¹/₃ x ⁵/₄ = ⁵/₂₀ = ¹/₄
What are iterative methods? (➐)
Iterative methods are techniques where you keep repeating a calculation in order to get closer and closer to the solution you want. You usually put the value you've just found back into the calculation to find a better value. You can use these when an equation is too hard to solve.
What is the LCM? (➄)
LCM means Lowest Common Multiple and it is the smallest number that will divide by all the numbers in question. E.G. The LCM of 3 and 5 is 15 as it is the smallest multiple that is common between these two numbers.
What are multiples and how can they be found? (➂)
Multiples are just the times table of a number. Multiples of a number can be found by multiplying it by another number. E.G. multiples of 2 are: 2,4,6,8...
How do you prove that (n + 3)² - (n - 2)² ≣ 5(2n +1)? (➄)
Take one side of the equation and play about with it until you get the other side: • LHS: (n + 3)² - (n - 2)² ≣ n² + 6n + 9 - (n² - 4n + 4). • ≣ n² + 6n + 9 - n² + 4n - 4. • ≣ 10n + 5. • ≣ 5(2n + 1) = RHS ('≣' means identically equal.)
What are prime numbers? (➂)
A prime number is a number which doesn't divide by anything, apart from itself and one. The only exception is 1, which is NOT a prime number.
How can you find the reciprocal of a number? (➂)
By taking the number and dividing 1 by the number. E.G. the reciprocal of 8 is 1/8.
How do you multiply fractions? (➂)
Cancel or reduce if possible, then multiply straight across. Some people like to multiply straight across and then reduce. Either way, if you can reduce, you must do so. E.G: ¾ x ½ = ³/₈. In this example, the fraction doesn't simplify.
What are factors and how can they be found? (➂)
The factors of a number are all numbers that divide into it. There's a method that guarantees you will find them all: 1) Start by multiplying 1 and the number, then try it with 2, then 3 and so on, listing the pairs in rows. 2) Try each one and cross out a row if it doesn't divide exactly. 3) Eventually you will get a row that's repeated, at which point you should stop as you now have all the factors of that number.
What do the inequality symbols mean? (➂)
• < means Less than. • > means Greater than. • ≤ means Less than or equal to. • ≥ means Greater than or equal to.
How do you turn fractions into recurring decimals? (➐)
• Write ⁸/₃₃ as a recurring decimal. 1) Find an equivalent fraction with all nines on the bottom. The number on the top will tell you the recurring part. 2) You can also use long division.
What are significant figures? (➂)
The first significant figure of any number is simply the first digit which isn't a zero (left to right). Then after the first, the second, third etc. figures can be anything (including zeroes).
What are the rules of standard form? (➃)
The standard format is: A x 10ⁿ 1) The front number is always between 1 and 10. 2) The power of 10, ⁿ, is how far the decimal point moves. 3) ⁿ is positive for big numbers, ⁿ is negative for small numbers.
How do you cancel down fractions? (➂)
To cancel down or simplify a fraction, divide the top and bottom by the same number until they won't go further. E.G: Simplify ¹⁸/₃₄: • ¹⁸/₃₄ = ⁶/₈ (dividing top and bottom by 3). • ⁶/₈ = 3/4 (dividing by 2). This wont go further and so this is the answer.
How can you estimate calculations? (➃)
To estimate, you simply round all the numbers to easier ones (1 or 2 significant figures generally does the trick). You can then round again to make further steps easier if you need to. E.G: (127.8 + 41.9)/(56.5 x 3.2) ≈ (130 + 40)/(60 x 3) = 170/180 ≈ 1
How do you round to significant figures? (➃)
To locate the the last digit, you have to find the first figure (from left to right) that isn't a zero. Then, you find the decider again and round up or down depending on whether or not it is greater than 5.
What is the inequalities rule for multiplying by negative numbers? (➄)
When multiplying or dividing by a negative number, the inequality is reversed (basically if x changes sides, the inequality changes).
What are the rules for multiplying or dividing with negatives? (➂)
1) + and + = + 2) + and - = - 3) - and + = - 4) - and - = + 5) Examples: -2 x 3 = -6, -8 ÷ -2 = +4, x + -y - -z = x - y + z.
What's the method to factorise quadratics if a=1? (➄)
1) Always rearrange into the standard into the standard format: x² + bx + c = 0. 2) Write down the two brackets with the x's in: (x )(x ) = 0. 3) Then find 2 numbers that multiply to give 'c' (the end number) but also add/subtract to give 'b' (the coefficient of x). 4) Fill in the + or - signs and make sure they workout properly. 5) As an essential check, expand the brackets to make sure they give the original equation. 6) Finally, solve the equation by setting each bracket equal to 0.
How can we disprove things mathematically? (➃)
1) Disprove things by finding a counter example. 2) WORKED EXAMPLE: • Ross says "the difference between any two consecutive square numbers is always a prime number." Prove that Ross is wrong. • Just keep trying pairs of consecutive square numbers (e.g. 1² and 2²) until you found one that doesn't work. • 1 and 4 - difference = 3 (a prime number) • 4 and 9 - difference = 5 (a prime number) • 9 and 16 - difference = 7 (a prime number) • 16 and 25 - difference = 9 (NOT a prime number). • Therefore, Ross is incorrect.
How can you solve equations? (➂)
1) First, rearrange the equation so that all the x's are on one side and the numbers are on the other. Combine terms where you can. 2) Then divide both sides by the number multiplying x to find the value of x. 3) WORKED EXAMPLE: • Solve 5x + 4 = 8x - 5 • 5x + 9 = 8x (we took the -5 on the right side and added it to +4 on the left side.) • 9 = 3x (we took the +5x on the left side and subtracted it from the +8x on the right side. We now have a number value on one side and a value with 'x' on the other.) • x = 3 (we divided 9 by the '3' in '3x' so that we only have 'x' left on one side. 9 ÷ 3 = 3, so x=3.)
How does the completed square help you sketch a graph? (➒)
1) For a positive graph (where the coefficient is positive), the adjusting number tells you the minimum y-value of the graph. If the completed square is a(x + m)² + n, the minimum y value will occur when the bracket are equal to 0 (because the bit in brackets is squared, so is never negative). 2) The solutions to the equation tell you where the graph crosses the x-axis as it will always be greater than 0 (this means that the quadratic has no real roots). 3) If the adjusting number is positive, the graph will never cross the x-axis.
How do you add or subtract numbers in standard form? (➄)
1) Make sure the powers of 10 are the same - you'll probably need to rewrite one of them. 2) Add or subtract the front numbers. 3) Convert the answer to standard form if necessary. 4) Worked example: • Calculate (9.8 x 10⁴) + (6.6 x 10³) without using a calculator. Give your answer in standard form. • Rewrite one number so both powers of 10 are equal: (9.8 x 10⁴) + (0.66 x 10⁴). • Now add the front numbers: (9.8 + 0.66) x 10⁴ = 10.46 x 10⁴ • = 1.04 x 10⁵
What are the three harder rules of powers? (➐)
1) Negative powers rule: these need to be turned upside down. (Example one: 7-² = 1/7² = 1/49.) (Example two: (³/₅)-² = (⁵/₃)² = 5²/3² = ²⁵/₉.) 2) Fractional powers rule: • the power 1/2 means square root. (E.G: 25^¹/₂ = √25 = 5) • the power 1/3 means cube root. (E.G: 64^¹/₃ = ∛64 = 4) • the power 1/4 means fourth root. (E.G: 81^¹/₄ = ∜81 = 3) 3) Two stage fractional powers rule: with fractional powers like 64^⁵/₆, always split the fraction into a root and a power, and do them in that order, root first, then the power: • = (64)^¹/₆ x 5 • = (64^¹/₆)⁵ • = (2)⁵ • = 32
How do you simplify or collect like terms? (➂)
1) Put bubbles around each term - be sure to capture the +/- sign in front of each. 2) Then you can move the bubbles into the best order so that like terms are together. 3) Combine like terms.
What are the six steps to solve easier simultaneous equations? (➄)
1) Rearrange both equations into the form: ax + by = c, and label them '1' and '2'. 2) Match up the numbers in the front (the coefficients) of either the x's or the y's in both equations. You may need to multiply one or both equations by a suitable number. Relabel them '3' and '4'. 3) Add or subtract the two equations to eliminate the terms with the same coefficient. 4) Solve the resulting equation. 5) Substitute the value you've found back into equation '1' and solve it. 6) Substitute both of these values into equation '2' to make sure it works. If it doesn't, go back and check your working - you've made a mistake.
What are the seven steps to solve harder simultaneous equations? (➑)
1) Rearrange the quadratic equation so that you have the non-quadratic unknown on its own. Label the two equations '1' and '2'. 2) Substitute the quadratic expression into the other equation. You'll get another equation - label it '3'. 3) Rearrange to a quadratic equation and solve it. 4) Stick the first value back in one of the original equations (pick the easy one). 5) Stick the second value back in the same original equation to check they work. 6) Substitute both pairs of answers back into the other original equations to check they work. 7) Write the pairs of answers out again, clearly at the bottom of your working.
How do you multiply or divide numbers in standard form? (➄)
1) Rearrange to put the front numbers and the powers of 10 together, 2) Multiply or divide the front numbers, and use the power rules to multiply or divide the powers of 10. 3) Make sure you answer is still in standard form. 4) Worked example: • Find (2 x 10²) x (6.75 x 10²) without using a calculator. Give your answer in standard form. • (2 x 10²) x (6.75 x 10²) = (2 x 6.75) x (10² x 10²) • 13.5 x 10² + 2 = 13.5 x 10⁴ • 1.35 x 10 x 10⁴ • 1.35 x 10⁵
How do you factorise brackets? (➃)
1) Take out the biggest number that goes into all the terms. 2) For each letter in turn, take out the highest power that will go into every number. 3) Open the bracket and fill in all the bits needed to reproduce each term. 4) Check your answer by multiplying out the bracket and making sure it matches the original expression. 5) Worked example: • Factorise 3x² + 6x. • The biggest number that divides into 3 and 6 is '3' and the highest power that goes into both is 'x'. • The expression is therefore: 3x(x+2).
What is the difference of two squares? (➅)
1) This is where you have 'one thing squared' take away 'another thing squared'. 2) There's a quick and easy way to factorise it, you use the rule: a² - b² = (a + b)(a² - b²). 3) Examples: • Factorise 9p² - 16q². • = (3p + 4q)(3p - 4q). • Here, you just had to spot that '9' and '16' are square numbers. • Factorise 3x² - 75y². • = 3(x² - 25y²). • = 3(x + 5y)(x - 5y). • Here, you had to take out a factor of 3 first. • Factorise x² - 5. • (x + √5)(x - √5) • Although 5 isn't a square number, you can write as (√5).
How do you multiply/divide algebraic fractions? (➑)
1) To multiply two fractions, multiply tops and bottoms separately and to divide, turn the second fraction upside down and then multiply. 2) WORKED EXAMPLE: • Simplify: x² - 4/x² + x - 12 ÷ 2x + 4/x² - 3x. • Turn the second fraction upside down. • (x² - 4)/(x² + x - 12) x (x² - 3x)/2x + 4) • Factorise both fractions. • (x² - 4)/(x² + x - 12) = (x + 2)(x - 2)/(x + 4)(x - 3). • (x² - 3x)/2x + 4) = x(x - 3)/2(x + 2) • Cancel out both fractions. • (x + 2)(x - 2)/(x + 4)(x - 3) = x - 2/x + 4 • x(x - 3)/2(x + 2) = x/2 • You now have: (x-2)/(x+4) x (x)/(2) • Multiply top and bottom. • Multiplying the top gives: (x-2) x (x) = x(x-2). • Multiplying the bottom gives: (x+4) x (2) = 2(x+4). • This gives us: x(x-2)/(2(x+4)
How do you convert between fraction, decimals and percentages? (➂)
1) To turn fractions into decimals you have to divide the numerator by the denominator. 2) To turn decimals into percentages, you have to multiply by 100. 3) To turn percentages into decimals, you divide by a hundred. 4) Converting decimals into fractions varies, although, a normal decimal (one that isn't recurring) is multiplied by a multiple of 10 so that it becomes an integer and is turned into a fraction by putting the number you got over the denominator (which will be the multiple of ten you used to turn the decimal into an integer).
How can we tackle difficult questions with the iteration methods? (➐)
1) WORKED EXAMPLE: A solution to the equation x³ - 3x - 1 = 0 lies between -1 and -2. By considering values in this interval, find a solution to this equation to 1 d.p. • Try (in order) the values of c with 1 d.p. that lie between -1 and -2. There's a sign change between -1.5 and -1.6, so the solution lies in this interval. • Now try values of x with 2 d.p. between -1.5 and -1.6. There's a sign change between -1.53 ad -1.54, so the solution lies in this interval. • Both -1.53 and -1.54 round to -1.5 to 1 d.p. so a solution to x³ - 3x - 1 = 0 is x = -1.5 to 1 d.p.
How can you sketch a quadratic inequality to help you solve a quadratic inequality? (➒)
1) WORKED EXAMPLE: Solve the inequality: x² + 2x - 3 > 0. • Set the quadratic = 0 and factorise. • x² + 2x - 3 = 0 • (x + 3)(x - 1) = 0 • Now solve the equation to see where it crosses the axis : (x + 3)(x - 1) = 0 • (x + 3) = 0, so x = -3 • (x - 1) = 0, so x = 1. • Then sketch the graph - it'll cross the x-axis at -3 and 1, and because the x² term is positive, it'll be a U-shaped curve. • Now solve the inequality - you want the bit where the graph is above the x-axis (as it's >). Reading off the graph, you can see that it is -3 < x < 1.
How can you use sequences to solve problems? (➅)
1) WORKED EXAMPLE: The nth term of a sequence is given by the expression 4n - 5. The sum of two consecutive terms is 186. Find the value of the two terms. • Call the two terms you are looking for n and n+1. • Then, their sum is: 4n - 5 + 4(n + 1) - 5 • = 4n - 5 + 4n + 4 - 5 • = 8n - 6. • Now we simply set it equal to 186 and solve the equation: 8n - 6 = 186. • 8n = 192 • n = 24. • This means that you need to find the 24th and 25th terms.: • n = 24: (4 x 24) - 5 = 96 - 5 = 91 • n = 25: (4 x 25) - 5 = 100 - 5 = 95 • The next two terms are therefore: 91 and 95.
How can you show inequalities on a graph? (➅)
1) WORKED EXAMPLE: Shade the region that satisfies y ≤ x + 2. • Convert the it into an equation: y = x + 2. • Workout which side of the line you want. Put x = 0 and y = 0 (the origin) into each equality and see if it makes the inequality true or false. • y ≤ x + 2 ➠ 0 ≤ 0 + 2 ➠ 0 ≤ 2 • 0 is indeed less than 2. Although it isn't equal to 2, this is still counted as true which means, the origin's on the correct side of the line and that part will be shaded.
How do you solve quadratics by completing the square when 'a' = 1? (➑)
1) WORKED EXAMPLE: • Express x² + 8x + 5 in the form (x + m)² + n. • x² + 8x + 5 - it's in the standard format. • Write out the initial bracket: (x = b/2)² = (x + 4)². • Multiply out the brackets & compare to original. • (x + 4)² = x² + 8x + 16 (original had +5 here). • (x + 4)² - 11 = x² + 8x + 16 - 11 (you need 16 - 5 = 11). • (x + 4)² - 11 = x² + 8x + 5 (matches the original). • Answer = (x + 4)² - 11.
How can you tackle typical nth term questions involving a quadratic sequence? (➐)
1) WORKED EXAMPLE: • Find an expression for the nth term of the sequence that starts 10, 14, 20, 28, ... • the n sequence: 1, 2, 3, 4 • quadratic term: 10, 14, 20, 28 • 1st difference: +4, +6, +8 • 2nd difference: +2, +2 • We have to divide the value of the second difference by 2, to get the coefficient of the n² term. Here, it's: 2 ÷ 2 = 1, so the coefficient is 1 and it will just be written as n². • quadratic term: 10, 14, 20, 28 • the n² sequence: 1, 4, 9, 16 • Term - n² sequence: 9, 10, 11, 12. • The nth term for this sequence is n+8. • The expression is therefore: n² + n + 8.
How do you tackle typical nth term questions involving linear sequences? (➃)
1) WORKED EXAMPLE: • Find an expression for the nth term of the sequence that starts: 5, 8, 11, 14, ... • n sequence: 1, 2, 3, 4 • linear term: 5, 8, 11, 14 • The common difference is 3 (the linear term goes up by 3 each time), so '3n' is in the formula. • '3n' term: 3, 6, 9, 12 • first term: 5, 8, 11, 14 • To get from the '3n' term to the original term, you have to add 2 each time (3 + 2 = 5, 6 + 2 = 8, etc.). • The expression for this term is therefore: 3n + 2.
How can you use the 7-step method to rearrange formulas when the subject is in a fraction? (➄)
1) WORKED EXAMPLE: • Make b the subject of the formula: a = 5b + ¾ • There aren't any √ signs so ignore step one. Get rid of the fraction by multiplying to get rid of the denominator. • 4a = 5b + 3 (multiplied 'a' by 4 - the denominator). • There aren't any brackets so ignore step 3. • Collect all the subject terms on one side and all non-subject terms on the other. • 5b = 4a - 3 • It's now in the form 'Ab = B' (where A = 5 and B = 4a - 3). • Divide both sides by 5 to give 'b = '. • b = 4a - ⅗.
How can you use the 7-step method to rearrange formulas when the subject appears twice? (➅)
1) WORKED EXAMPLE: • Make p the subject of the formula: q = p + 1/ p -1 • There aren't any √ so ignore step 1. • Get rid of any fractions. • q(p - 1) = p + 1 • Multiply out the brackets. • pq - q = p + 1 • Collect all the subject terms on one side and all the non-subject terms on the other side. • pq - p = q + 1 • Combine like terms on each side of the equation. • p(q - 1) = q + 1 • Divide both sides by (q - 1) to give 'p = '. • p = q + 1/q - 1.
What is an integer? (➂)
An integer is another name for a whole number - either a positive or negative number or zero.
How can you use the 7-step method to rearrange formulas when the subject is squared? (➄)
1) WORKED EXAMPLE: • Make u the subject of the formula: v² = u² + 2as. • There aren't any √, fractions or brackets so ignore steps 1-3. • Collect all the subject terms on one side and all the non-subject terms on the other. • u² = v² - 2as. • It's now in the form: Au² = B (where A = 1 and B = v² - 2as). • A = 1 means it's already in the form 'u² = '. • Square root both sides to get 'u = ± '. • u = ±√v² - 2as
How do you simplify algebraic fractions? (➅)
1) WORKED EXAMPLE: • Simplify 21x³y²/14xy³. • Divide by 7 on the top and bottom, then divide by x on top and bottom to leave x² on the top, then divide by y² on the top and bottom to leave y on the bottom. • = 3x²/2y
How do you factorise a quadratic when a ≠ 1? (➐)
1) WORKED EXAMPLE: • Solve: 3x² + 7x - 6 = 0 • (3x )(x ) = 0 • Number pairs: 1 x 6 and 2 x 3. • Find the pair that multiplies to give 'c' (6) & adds to give 'b' (7). Try each pair in 2 positions. • (3x 1)(x 6) = multiplies to give 18x & 1x, which add/subtract to give 17x & 19x. ☒ • (3x 6)(x 1) = multiplies to give 3x & 6x, which add/subtract to give 9x & 3x. ☒ • (3x 3)(x 2) = multiplies to give 6x & 3x, which add/subtract to give 9x & 3x. ☒ • (3x 2)(x 3) = multiplies to give 9x & 2x, which add/subtract to give 11x & 7x. ☑ • (3x 2)(x 3) = 0 (Fill in the initial brackets). • (3x - 2)(x + 3) = 0 (Fill in the signs). • (3x - 2) = 0 ➠ 3x = 2 ➠ x = ⅔ • (x + 3) = 0 ➠ x = -3 • ✬ Answer ➠ x = ⅔ OR x = -3 ✬
How do you factorise a quadratic when a = 1? (➄)
1) WORKED EXAMPLE: • Solve: x² - x = 12. • (x )(x ) = 0 • Find the right pairs of numbers that multiply to give c (12) and add or subtract to give b (1). Remember that the signs aren't important right now. In this case, the numbers are 3 and 4. • Now you have to fill in the gaps so that 3 and 4 add/subtract to give b (-1). • (x 3)(x 4) = 0 • (x + 3)(x - 4) = 0 • Make sure that the brackets expand to give the original expression. • Solve the equation. • (x + 3) = 0 ➠ x = -3 • (x - 4) = 0 ➠ x = 4 • Answer ➠ x = -3 OR x = 4.
How do you solve quadratics by completing the square when 'a' ≠ 1? (➒)
1) WORKED EXAMPLE: • Write 2x² + 5x + 9 in the form a(x + m)² + n. • 2x² + 5x + 9 - it's in the standard format. • Take out a factor of 2: 2(x + 5/2x) + 9. • Write out the initial bracket: 2(x + 5/2a)² = 2(x + 5/4)². • Multiply out the brackets & compare to original. • 2(x + 5/4)² = 2x² + 5x + 25/8 (original had +9 here). • 2(x + 5/4)² + 47/8 = 2x² + 5x + 25/8 + 47/8 (you need 9 - 25/8 = 47/8). • 2(x + 5/4)² + 47/8 = 2x² + 5x + 25/8 + 47/8. • Answer = 2(x + 5/4)² + 47/8.
How do you find maximum and minimum values for calculations? (➐)
1) When a calculation is done using rounded values, there will be discrepancy between the calculated value and the actual value. 2) Worked example: A pinboard is measured as being 0.89 m wide and 1.23 m long, to the nearest cm. A) Calculate the minimum possible values for the area of the pinboard. • Find the values for the bounds for the width and length: 0.885 m ≤ width < 0.895 m & 1.225 m ≤ length < 1.235 m. • Find the minimum area by multiplying the lower bounds and the maximum by multiplying the upper bounds: minimum area = 0.885 x 1.225 = 1.084125 m², maximum area = 0.895 x 1.235 = 1.105325 m². B) Use your answer to part A) to give the area of the pinboard to an appropriate degree of accuracy. • The area of the pinboard lies in the interval 1.084125 m² ≤ a < 1.105325 m². Both the upper bound and the lower bound round to 1.1 m² (1 d.p.), so area of the pinboard = 1.1 m² (1 d.p.).
What are the basic rules of algebra? (➂)
1) abc means a x b x c. 2) gn² means g x n x n. 3) (gn)² means g x g x n x n. 4) p(q-r)² means p x (q-r) x (q-r). 5) -3² is ambiguous. It should be written as either (-3)² = 9 or, -(3²) = -9, but you would usually take -3² to be -9.
What are the seven rules for powers and roots? (➄)
1) When multiplying, you add the powers (e.g: 4² + 4² = 4⁴). 2) When dividing, you subtract the powers (e.g: 4⁵ ÷ 4³ = 4²). 3) When raising the power from one power to another, you multiply the powers (e.g: (3²)² = 3⁴). 4) x¹ = x. Anything to the power of one is just itself. 5) x⁰ = 1. Anything to the power of zero is one. 6) 1ⁿ = 1. One to any power is just one.. 7) Fractions - apply the power to the top and bottom (e.g: (1⅗)² = (8/5)² = 8²/5² = ⁶⁴/₂₅).
What are five crucial details about the quadratic formula? (➐)
1) Whenever you get a minus sign be careful. If either 'a' or 'c' are negative, -4ac becomes +4ac. Also be careful if b is negative, as -b will be positive. 2) Remember it is '2a' on the bottom, not just 'a' - and you divide all of the top by '2a'. 3) The ± means you will get two solutions (by replacing it in the final step with '+' and '-'.) 4) If you get a negative number inside your square root, check your working as this will not be correct. 5) You should only use it if you: have a quadratic that won't easily factorise, the question mentions decimal places or significant figures, if the question asks for exact answers or surds.
How do you add/subtract algebraic fractions? (➑)
1) Work out a common denominator, then multiply top and bottom of each fraction by whatever gives you the common denominator. Add or subtract numerators only. 2) WORKED EXAMPLE: • Write 3/(x+3) + 1(x-2) as a single fraction. • Multiply the top and bottom of the first fraction by (x-2) and multiply the top and bottom of the second fraction by (x+3). • = 3(x-2)/(x+3)(x-2) + (x+3)/(x+3)(x-2) • = 3x-6/(x+3)(x-2) + x+3/(x+3)(x-2) • = 4x - 3/(x+3)(x-2)
What is a rational number? (➄)
A number that can be written as a fraction. They can come in three different forms, as either: integers (e.g. 4 = 4/1), fractions where p and q are non zero integers (e.g. 1/2), terminating/recurring decimals (e.g. 0.3 recurring).
What are the 6 rules for manipulating surds? (➐)
1) √a x √b = √a x b • e.g. √2 x √3 = √6 • e.g. (√b)² = √b x √b = √b x b = b 2) √a/√b = √a/b • e.g. √8/√2 = √8/2 = √4 = 2 3) √a + √b - Do nothing! It definitely isn't √a + b 4) (√a + √b)² = (√a + √b)(√a + √b) = a² + 2a√b + b 5) (√a + √b)(√a - √b) = a² + a√b - a√b - (√b)² = a² - b 6) a/√b = a/√b x √b/√b = a√b/b - This is known as rationalising the denominator, where you get rid of the √ on the bottom of the fraction. For denominators of the form a ± √b, you always multiply by the denominator but change the sign in front of the root.
How do you add or subtract fractions? (➃)
Find a common denominator and make equivalent fractions. Then add or subtract.
How can we find the LCM of numbers? (➄)
If we already know the prime factors of the numbers, you can use this method: 1) List all the prime factors that appear in either number. 2) If a factor appears more than once in one of the numbers, list it that many times. 3) Multiply these together to give the LCM. 4) WORKED EXAMPLE: • 18 = 2 x 3² and 30 = 2 x 3 x 5. • 18 = 2 x 3 x 3 and 30 = 2 x 3 x 5 • So the prime factors that appear in either numbers are: 2, 3, 3, 5 (three is listed twice as it appears twice in 18). • LCM = 2 x 3 x 3 x 5 = 90
How do you convert a mixed number into an improper fraction? (➂)
Mixed numbers are things such as 3½, with an integer part and a fraction part. Improper fractions are ones where the top number is larger than the bottom number. Here is a worked example of how to convert a mixed number to an improper (or top heavy) fraction: • Write 4²/₃ as an improper fraction. 1) Think of a mixed number as an addition: 4 + ²/₃. 2) Turn the integer part into a fraction: 4 + ²/₃ = ¹²/₃ + ²/₃ = ¹⁴/₃
How do you multiply out single brackets? (➂)
The main thing to remember when multiplying out brackets is that the thing outside the bracket multiplies each separate term inside the bracket.
How can you simply expression using the 6 rules of manipulating surds? (➑)
Worked examples: 1) Write √300 + √48 - 2√75 in the form a√3, where a is an integer. • Write each surd in terms of √3: a) √300 = √100 x 3 = √100 x √3 = 10√3 b) √48 = √16 x 3 = √16 x √3 = 4√3 c) 2√75 = 2√25 x 3 = 2 x √25 x √3 = 10√3 • Then do the sum (leaving your answer in terms of 3): 10√3 + 4√3 - 10√3 = 4√3 2) Write 3/2 + √5 in the form a + b√5, where a and b are integers. • To rationalise the denominator, multiply the top and bottom by 2 - √5: a) 3/2 + √5 = 3(2 - √5)/(2+√5) - (2 - √5) b) = 6 - 3√5/2² - 2√5 + 2√5 - (√5)² c) = 6 - 3√5/4 -5 d) = 6 - 3√5/-1 e) = -6 + 3√5 (a = -6 and b = 3)
How can you find prime factors? (➃)
• Numbers can be broken down into a string of prime factors all multiplied together, called 'prime factor decomposition' or 'prime factorisation'. • Start with the number at the top, and split it into two factors, writing these two factors on two separate, individual branches below the number. Every time you get a prime number, you circle it. • Keep going until you can't go further (until you are left with just primes), then just write out all the primes in order. If there's more than one of the same factor, you can write them as powers.
