Linear and Quadratic Equations

If ab(a + b) ≠ 0, then (a^4 b + a^2 b^3 + ​a^3 b^2 + a b^4) / ab(a+b)=?

yes, we can switch terms since they sign is positive. i tested different scenarios in excel and the result is the same for all types of a and b.

If the product of two integers is 1, then either both are...

...1 or both are -1 EXAMPLES If x and y are integers, what is the value of x + y? 1) x(xy)=1 2) x/y=1 If x and y are integers, what is the value of x + y? 1) x(x^2 y)=1 2) y(y^2 x)=1

Question 4 - If x^3−x^2−4x+4=0, what is the sum of all possible values of x? 0 1 2 3 4

1 this is a recurring mistake of mine. i need to make sure each step i am takign to simplify the equation. in the second term, i factored out 4 instead of -4

Mark purchased a number of brownies and cookies from the school cafeteria. Each cookie costs $1.10, and each brownie $0.80. If Mark spent $9.90 on the brownies and cookies, how many of the brownies did he purchase? 1 3 7 11 13

Can approach with smart numbers or algebra. below is the explanation of algebraic approach in the image: ((Since cookies and brownies must be purchased in whole numbers, 9 - c must be divisible by 8. The only positive integer value of c that will allow 9 - c to be divisible by 8 is 1. (A value of 9 for c would also make this equation true, but both cookies and brownies were purchased so b can't be zero.) Therefore 1 cookie was purchased, which is enough information to determine the quantity of brownies purchased.)) smart numbers: - dont be afraid of manipulating the numbers to make them sound easier. so multiply all numbers by 10 to have 11, 99, 8. - now, try out smart numbers of the easiest to do operations with. between 8 and 11, 11 is much easier to try out. so start with it. - start with a large one. you know the number has to be less than 7.

What is the value of d? 1) n^2+an+b=(n+c)(n+d) for all values of n. 2) a - c = 10

Look carefully in the first equation...see what the test makers are tricking yo not to see...the right side does not have + in between the parentheses. it is all one term. so you cant factor out. ⇒n2+an+b=(n+c)(n+d) for all values of n. It must be true that c and d add to produce a and multiply to produce b. This gives us two subtle, yet important equations. First c + d = a. and second cd = b. We can isolate d in the first equation: d = a - c, but this is not enough information to answer the question. Therefore, statement one alone is not sufficient. If you are unsure where c + d = a and cd = b came from, consider the following. Let's take a simpler quadratic that has real numbers: n^2+8n+16=(n+4)(n+4). Notice that 4 + 4 = 8 and 4 × 4 = 16.

If p, q, y and z are integers, and y(y^2 z)=1, is y^3×z^4=pq? 1) p = 4 2) q = 4

Question Stem Analysis: Since y and z are integers, and y(y^2 z)=1 →y^3 z=1, it must be that y and z are both 1 or are both -1. For example, 13 × 1 = 1, and (-1)3 × (-1) = 1. We must determine whether y^3×z^4=pq. Remember that p and q are integers. Statement One Alone: ⇒p = 4 From the stem, we know that y and z are both 1 or are both -1. Thus, y^3×z^4 is either 1 or -1 depending on whether y is 1 or -1. Statement one tells us that p = 4. Since the stem tells us that both p and q are integers, pq cannot be 1 or -1, and thus y^3×z^4≠pq. Statement one alone is sufficient. Eliminate answer choices B, C, and E. Statement Two Alone: ⇒ q = 4 From the stem, we know that y and z are both 1 or are both -1. Thus, y^3×z^4 is either 1 or -1 depending on whether y is 1 or -1. Statement two tells us that q = 4. Since the stem tells us that both p and q are integers, pq cannot be 1 or -1, and thus y^3×z^4≠pq. Statement two alone is sufficient.

If ab - b + (a^0 - a^2) ≠ 0, (a−b+1) / ab−b+(a^0−a^2) =

This problem combines so many tricky algebraic tricks. it is best to review it on desktop

When to divide or multiply by a variable?

When the question explicitly says that the variable is neither a zero, nor, expression of zero. x/1+y question will say y<>-1 , meaning that the whole expression will not equal zero. but sure, maybe y is zero, then when multiplying by (1+y) we have an expression of 1, not zero.

Question 4 - Which of the following is not a difference of two squares? 25 - x^4 25 - x^9 25 - x^16 25 - x^36 25 - x^64

b Binomials in which one or both terms contain a variable with an odd exponent cannot be a difference of two squares: the square of a term must result in an even exponent.

Question 12 - so neat What is the value of x^2- 2x - 14? 1) x < 0 2) (x−5)(x+3)=0

b Statement Two Alone: ⇒(x−5)(x+3)=0 ⇒(x−5)(x+3)=0 ⇒x^2−2x −15 = 0 If x^2- 2x - 15 = 0, it must be true that x^2- 2x - 14 = 1, and statement two alone is sufficient.

Question 6 - "rewriting the q stem is everything" If x and y are integers, does 2xy = x^2 + y^2? 1) x + y = 4 2) (x - y)^2 = 0

b We know that x and y are integers. The question can be simplified. ⇒2xy=x^2+y^2? ⇒x^2+y^2−2xy=0? ⇒(x−y)^2=0?

Question 12 - "rewrite the q stem" If x and y are integers and y^2(1 + x) = 1, what is the value of y? 1) x = 0 2) y^999 ≠ 1

b from the q stem; The givens are that x and y are integers and that y^2(1 + x) = 1. Since x and y must be integers, the only way for y^2(1 + x) to equal 1 is for y^2 = 1 and for 1 + x = 1. Thus, x is 0, and y is positive or negative 1. We need to determine the value of y.

Question 7 - 'factor out until you die" If x ≠ - 2, then 2x^4−32 / 2x+4 is equal to which of the following? x - 2 x + 4 x^2 + 4 (x - 2)(x^2 + 4) 2x - 8

d factor out until you die

Question 1 - "complicated factoring out" If the sum of a number and its reciprocal is 5/2, which of the following can be the value of this number? 1/5 2/5 1 2 5

d fast solution is to try out the options. but a complicated factoring out must be learned here after reaching ⇒2x^2−5x+2=0, we will need to determine the constants. When determining the constants, we need to find two numbers that multiply to 2 and also two numbers that sum to -5 WHEN ONE OF THOSE NUMBERS IS MULTIPLIED BY 2. This is because the 2x^2 term will break down to x and 2x. We see that the factors of 2 are: 1 and 2 or -1 and -2 We also see that when we multiply -2 by 2 we get -4 and -4 - 1 = -5. Thus, it becomes clear that the factored version of the quadratic is: ⇒(2x−1)(x−2)=0⇒x=1/2 or x=2

Question 10 - "same fundamental principles applied" If x^4−13x^2+36=0, what is the sum of all possible values of x? 4 3 2 1 0

e We can explore this further by factoring the more familiar quadratic equation x^2−13x+36=0. we need to find two numbers that sum to -13 and that multiply to 36. Those two numbers are -9 and -4, so we have: x^2−13x+36=0 (x−9)(x−4)=0 x=9 or x=4 Now, when we factor x^4−13x^2+36=0, we are again looking for two numbers that sum to -13 and multiply to 36, but the only change is that inside our parentheses, we replace x with x^2. Thus, the factored form is: x^4−13x^2+36=0 (x^2−9)(x^2−4)=0 Next, using the difference of two squares, a^2 - b^2 = (a + b)(a - b), we can simplify to obtain: (x−3)(x+3)(x−2)(x+2)=0 x=3, -3, 2, -2

999^2 − 1 is approximately what percent of 99^2 − 1? 100,009 10,183 1,003 101 11

great example of estimation

What is the value of s? 1) 1/s = 1/(s+25) + 1/(s+1) 2) −1/5 is not a reciprocal of s.

great example showing complicated use case for LCD

If (x+2)(6+ (3)/x)=0 and x ≠ -2, then x =? −1/2 −1/3 −1/4 1/4 1/2

it is all about that =0 in the question even though the question didn't specify x<>0 i tried to solve this Q twice and failed. this is a really important simple point.

3^10 − 2^10 is a multiple of all of the following numbers except: 5 25 55 75 211

it is so freaking tricky. i knew that prime factorization would help here. but i thought knowing the last digit of the result will work by the divisibility rules. i was wrong.

If r and t are nonzero and (4+s)/t + s/r = 1, which of the following properly expresses s in terms of t and r?

multiplying by variables since they are defined in the question..

If 2t + 4u = v and 2t - 4u = w, then t is equal to which of the following? (v+4)/w 4/(v+w) (v+w)/4 v + w 4v + w

no need to complicate things..

If (x + 9) is a factor of the expression x^2 - nx - 36, where n is constant, what is the value of n? -9 -5 -4 4 5

requires elite understanding of what does it mean to be a factor vs root

Contemplate the difference between the following questions on this topic; PS problem: If (x + 9) is a factor of the expression x^2 - nx - 36, where n is constant, what is the value of n? If x^2- kx = 16, what is the value of k + x? 1) (x + 4) is a factor of x^2- kx - 16 2) x > 0 If x^2 + kx = -16, what is the value of k + x? 1) (x + 4) is a factor of x^2 + kx + 16 2) x=√−16−kx (radical for all terms)

the depth of the solution varies in each one. will let you contemplate on this... source: TTP 6th M test, Q13 2nd H test, Q12 3rd H test, Q11

If a ≠ 3, (a^2 + 12) / (a−3) + 7a / (3−a) = a - 3 a - 4 a a + 3 a + 4

think neatly and creatively. i should have pushed my brain to think creatively. this is the purpose of deliberate practice. my mind is not usually trained to look this creatively.