Algebra 1
Special Binomial Multiplication
(4x-3)² (5x+2)(5x-2) (4x-3)(4x-3) 25x²-10x+10x-4 16x²-12x-12x+9 25x²-4 16x²-24x+9
Associative Property of Addition
(a+b)+c = a+(b+c)
Associative Property of Multiplication
(ab)c = a(bc)
Multiply in Binomial
(x+5)(x-1) x²-1x+5x-5 x²+4x-5
7a-4(b+2) if; a=5 b=2
19
3x+2x-x+2x^2
2x^2+4x
(6+25-7)÷6
4
Inverse Variation
A negative or positive constant rate linear function that avoids the origins. Equation is y=k/x (k = yx) is the constant of the proportionality. Ex. y=3/x or y=-2/x.
Function
A relation is a function if and only if the element in the domain is paired with a unique output.
f(x)=1/2x+3
f(x)=1/2x+3; plug in any number in for x and solve. What you plug in for x is your Domain, and what you get for your y is your Range.
One solutions
two lines that intersect at one point
Which one is a Function? { (5,6)(3,2)(-3,2)(4,1) } { (5,6)(1,4)(3,4)(1,-2) }
{ (5,6)(3,2)(-3,2)(4,1) }
√392
√196*√2=14√2
Distributing radicals
√5(√10+4√3= √50+4√15=√25√2+4√25= 5√2+4√15
4y^2-2y+6y^3-16y-10y^2+8y^3
+8y^3+4y^2-16y+6y^3-10y^2-2y= 14y^3-6y^2-18y
-4(8x-4)+3x
-29x+16
-(1-5n)-7n
-2n-1
Elimination Method
-8x-10y=28 1. Line up term 4x+10y=-24 2. Opposite coefficients -4x=4 3. Add each term -4x/-4=4/-4 4. Substitute into original equation x=-1 -8x-10y=28 (-1.-2) -8(-1)-10y=28 8-10y=28 -8 -8 -10y=20 -10y/-10=20/-10 y=-2
x⁰
1
Cubed Roots (Perfect Cubes)
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859,8000, 9261, 10648, 12167, 13824, 15,625. (This 1-25).
2/3x=24
1. (3/2) 2/3x=24(3/2) 2. x= 3(12) 3. x=36
(-20)(x+3/4)
1. (4)(-20)(x+3/4)(4) 2. -80=x+3 3. -83=x
Distributive Property (Algebraically)
1. 3(x+8) 2. 3x+3(8) 3. 3x+24
3x+5=38
1. 3x+5=38 -5 -5 2. 3x=33 3. 3x/3=33/3 4. x=11
Distributive Property (Numerically)
1. 7(53) 2. 7(50+3) 3. 7(50)+ 7(3) 4. 350+21 5. 371
a-3=10 (Addition Property)
1. a-3=10 +3 +3 2. a=13
x⁻¹
1/x¹
20÷(4-(10-8))
10
72÷9+7
15
Inequalities
2x+3(3-x) < 14+9 2x+9-3x < 23 -1x+9 < 23 -9 -9 -1x<14 -1x/-1 < 14/-1 x > -14
No Solution
2x+5(x-4)=7x-13 2x+5x-20=7x-13 7x-20=7x-13 +20 +20 7x=7x+7 -7x -7x 0=7
Factoring a polynomial #2
2x²+11x+12 Multiply the outside #´s Factors of 24 24x Bring down outside numbers 8,3 2x²+8x+3x+12 Find GCF of first two seperatly from 2x(x+4)+3(x+4) the last two terms Take the #'s not in ( ) and (2x+3)(x+4) put them in ( )
6x⁸/3x⁵
2x³
Infinite Soultions
3(2x-5)=x+2(2x-7)-1+x 6x-15=x+4x-14-1+x 6x-15=6x-15 +15 +15 6x=6x 6x/6=6x/6 x=x
Find the slope given two points; (4,-2)(-1,3)
3-(-2)/-1-4 5/-5=-1 slope: -1
Evaluating Expressions
3a+7=12 The whole thing is the equation, 3a+7 is the expression, the three is the Coefficient, the a is the variable, and 7 is the Constant.
Find the Slope given the equation; 3x-6y+8=0
3x-6y+8=0 -3x -3x -6y+8=-3x -8 -8 -6y=-3x-8 slope:1/2 -6y/-6=-3x-8/-6 y=1/2x+4/3
2x^2+2x+x^2+x
3x^3+3x
3√12
3√12=3√4*√3=3*2√3=6√3
8+7(7n-4)
49n-20
Determine if the point is a solution of the equation. 5x+2y=14 and (0,7)
5(0)+2(7)=14 0+14=14 14=14 true
Multi step Equation (Distributive Property)
5(4x-1)+2=24; first you need to distribute the 5 out to the 4x and the -1. Your equation will now look like this 20x-5+2=24. Next, you need to add and get rid of the constant. Your equation will now look like this 20x= 27. Then divide the coefficient and the 27 and then you'll have an answer of x=1.35.
One Solution
5x+2(3x-4)=8x+8 5x+6x-8=8x+8 11x-8=8x+8 11x=8x+16 3x=16 x=5 1/3
3x-2y+x^2 if; x=-3 and y=-4
8
Direct Variation
A negative or positive constant rate linear function that has a y-Intercept at (0,0). Equation is y=kx (k = y/x) is the constant of the proportionality. Ex. y=3x or y=-2.
Relations
A relationship between a set of values (ie. x-value and y-value).
Literal Equations
A/L = L*W/L;W (A)=(1/2bh);b A/L=W 2(A)= (2)(1/2bh) 2A=bh 2A/h=b
Operations with Radicals
Add or Subtract: 5√2+3√2-4√3+6√3= 8√2+2√3 √8+3√2+3√27+2√18=2√2+3√2+9√3+6√2= 11√2+9√3 Multiply or Divide: √5*√2= √10 1. √10*√15= 5*2 5*3 = 5√6 2. √14/√8 7*2 4*2 7 4=2√2= √7/2
Addition Symbol
Addition, add, sum, plus, more than, increased by. An examples a number more than 8 = 8+x.
Two Step Equation
An equation having two different operations.
Linear Inequality
An inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.
Cube Root
Cube Root examples 3√1=1 and 3√64=4
Is this Direct or Inverse Variation?; y=5x
Direct Variation
Division Symbol
Division, Divide, Divisible, Fraction, Quotient, Over, Each, Out of, Ratio. An example a quotient of a number and 7 = x/7
Factorial Symbol
Factorial [consecutive factors (products)]. An examples 6!=6×5×4×3×2×1= 720
A farm house shelters 11 animals. Some are goats and some are ducks. Altogether there are 34 legs. How many of each animal are there?
Figure out x and y. X= # of Goats legs =4 Y= # of Ducks legs = 2 Figure out first equation (The equation that takes the x and y and combines them into a #) x+y=11 Figure out the 2nd equation 4x+2y=34 Use the Substitution or the Elimination method to solve. Your answer should be (6 goats, 5 ducks)
Factoring a polynomial
Find factors that when multiplied equal the constant and when added equal the ¨bx¨. ax²+bx+c. The two inside numbers the ¨bx¨ should equal 3x. x²+3x+2 (x+1)(x+2) x²+1x+2x+2
GCF (Greatest Common Factor)
Finding the GCF of Polynomials. The biggest number that can be multiplied by a number to make the coefficient. Ex. 20a²+ 14a= 2a(10a+7) the 2a is the GCF.
Order of Operations GE(MD)(AS)
G: Grouping, E: Exponents, M: Multiply, D: Divide, A: Add, and S: Subtract. When on the stages of Multiplying, Dividing, Adding, and Subtracting always go from left to right.
Transitive Property of Equality
If a = b and b = c, then a = c
Substitution Property
If a = b+c and b = d then a = d+c
Addition Property of Equality
If a = b, then a+c = b+c
Subtraction Property of Equality
If a = b, then a-c = b-c
Multiplication Property of Equality
If a = b, then ac = bc
Horizontal Slope
If the slope is horizontal then the slope has a 0 as its rise and run. ex. 0/-3 = 0. What this mean is the two points have the same y-value so you would just start at the y-value and make a horizontal line. The slope is always 0.
Vertical Slope
If the slope is vertical then the slope has a 0 as its run. ex. -3/0 = ¨Error¨.What this mean is the two points have the same x-value so you would just start at the x-value and make a vertical line. The slope is always undefined.
Is this Direct or Inverse Variation?;y=5/x
Inverse Variation
Polynomials
Many Termed equation. When adding or subtract terms the exponents stays the same. Ex. 5y²+3y+2y²+6y=7y²+9y
Multiplication Symbol
Multiplication, Multiply, times, product of. An example the product of 6 and a number = 6x or 6(x)
Square Roots (Perfect Squares)
Numbers whose square roots are integers. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 566, 625. (This 1-25).
Monomial
One term ex. 2x
no solution
Parallel lines
Law of exponents
Product Rule: Coefficients are not exponents, Multiply coefficients ¨normally¨,Treat each base separately. Ex. a³ * a¹ = a⁴ Quotient Rule: Coefficients are not exponents, Simplify,reduce,divide coefficients(They are allowed to be negative). *Top heavy vs.Bottom Heavy* Ex. -15a⁶/-25a¹¹ = 3/5a⁵ Power Rule: Distribute the outside exponent to each base by multiplying. Even to the coefficients exponent. Ex. (4a⁶)² = 4²a¹²=16a¹² Combo: Ex. 3a²(2a¹)/4a⁸= 6a³/4a⁸=3/2a⁵
Combine Like Terms
Simplify expressions by adding or subtracting like terms. Key things to remember are like terms must have the same variable and they have to be in descending order.
2x+3y=6
Solving for y its 2x+3y=6 -2x -2x 3y=-2x+6 3y/3=-2x+6/ y=-2/3x+2
Square Root Symbol
Square root (x^2), Cubed root (x^3), radical, root. Some examples are √49=7, √121=11, √625=25, and 3√27=9.
Subtraction Symbol
Subtraction, subtract, minus less that decreased by. An examples 5 less than a number = x-5.
Slope using the equation
Take two points of the line. ex. (5,3) (6,1) Then put x_1 and y_1 on the first point and x_2 and y_2. Next, plug the numbers into the equation and the you'll have you're slope.
Absolute Value
The absolute value of a number is its distance from 0 on a number line. An example |-3|=3
Slope (rise/run)
The difference of the y-values divided by the difference of the x-values or the ratio vertical change and horizontal change.
Y=mx+b (Slope intercept form)
The m represents the slope and the b represents the y-intercept..
One Solution Parabola
The vertex is on one point and there are no other intersections.
Inverse Variation Table
The x-values divided by the constant is Inverse Variation. The y-value multiplied by the x-value is Inverse Variation.
Direct Variation Table
The x-values divided by the y-value is Direct Variation. The constant multiplied by the x-value is Direct Variation.
Domain
The x-values of a set - independent or input values. All the x-values have to be unique.
Range
The y-values of a set - dependent or output values.
Two solutions Parabola
There are to point of intersection.
No Solutions Parabola
This has no real solutions.
Trinomial
Three terms. 2x²+7x²+6y=9x²+6y
Infinite Solution
Two line over top of each other
Systems of Equations
Two or more equations that intersect at a single point. Ex. y=2x+5,y=1/2+8. Algebraically y=2x+5 -> 9=2(2)+5= 9=9 y=1/2x+8 -> 9=1/2(2)+8= 9=9.
Binomial
Two terms. ex. 2x+3x=5x
Function on a Graph
Use the line to see if it a function. Draw a vertical line down the graph and if it cross the line once it is a function. If it crosses more than once it is not a function.
Commutative Property of Addition
a + b + c = c + b + a
Reflexive Property
a = a
Line of Best Fit
a line drawn in a scatter plot to fit most of the dots and shows the relationship between the two sets of data
Distributive Property
a(b + c) = ab + ac
Multiplicative Property of Zero
a*0 = 0
Additive Indentity Property
a*1 = a
Additive Inverse Property
a+(-a)= 0
Multiplicative Inverse
a/b*b/a = 1 or 7*1/7=1
Communtative Property of Multiplication
abc = cba
Linear Equation
an equation between two variables that gives a straight line when plotted on a graph.
One Step Equation
an equation that requires only one step to solve...one operation
Quadratic Equations (Find the root)
ax²+bx+c=0 x²+11x+24=0 * Always equal 0* (x+8)(x+3)=0 Factor the Polynomial x+8=0 x+3=0 Set each factor for 0 solve for x -8 -3 x=-8 x=-3 These are the "Zeros""Roots""Solutions""x-Intercept"
Quadratic Formula
ax²+bx+c=0 and x = -b ± √(b² - 4ac)/2a. Ex. x²+7x+12=0 x= (-7)+-√7²-4(1)(12)/2(1) x=-7+-√49-48/2 x= -7+-√1/2 x=-7+-1/2 x=-7+1/2=-3 -3 and -4 are the roots x=-7-1/2=-4
Find the x-intercept and the y-intercept; -20x+40y=-80
for x-Inter.-20x+40y=-80 for y-Inter. -20(0)+40y=-80 40y=-80 -20x+40(0)=-80 40y/40=-80/40 -20x+0=-80 -20x=-80 y=-2 -20x/-20=-80/-20 x=4
Division Property of Equality
if a = b, then a/c = b/c
Symmetric Property of Equality
if a = b, then b = a
Simplifying Expression
to use distribution to combine like terms
Mapping Function x (5 3 -2) x (2 5 -1) x (0 -20 (Which Is A B C a Function?) y (4 5 0) y (3) y (4 6 7)
x (5 3 -2) x (2 5 -1) A B y (4 5 0) y (3)
Table (Which is a Function?) x y x y x y 4 5 2 2 5 -1 3 4 1 2 5 0 2 3 -2 2 5 1 1 4 0 2 5 2
x y x y 4 5 2 2 3 4 1 2 2 3 -2 2 1 4 0 2
Linear Equations Tables
x y y= -x+2 -2 4 -1 3 0 2 0-3/2-(-1)= -3/3 slope: -1 1 1 2 0
√x⁷
x³√x Always divide exponents by 2 2/7= 2*3=6 7-6=1 the number that had to be multiplied to get another number that isn't 2 is the exponent that goes outside. When you subtract the two numbers what left of on the inside of the root. If the is a 0 don't put anything.
(x⁵y⁷)²
x¹⁰y¹⁴
(x²)(x³)(x⁴)
x⁹
Point Slope form
y-y_1=m(x-x_1)
Substitution Method
y=(-2x+1) 2x-2(-2x+1)=4 y=-2(1)+1 2x-2y=4 2x+4x-2=4 y = -2+1 6x-2=4 y=-1 +2 +2 (1,-1) 6x=6 6x/6=6/6 x=1
Function Notation
y=2x+5 -> f(x)=2x=5 x y 3 11 4 3 5 15