If x is positive and y is 1 more than the square of x, then what is the value of x in terms of y ?
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Solution
Plug in x = 4, so y = 17. Now plug in 17 into the answers to see which gives you 4. Only choice (D) does.
The solutions of x2 + x – 20 = 0 are
Indicate all possible values.
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Solution
A and E
A and E Factoring this quadratic equation gives you (x + 5)(x – 4) = 0. For the first solution, x + 5 = 0, or x = –5. For the second solution, x – 4 = 0, or x = 4. Alternatively, you can PITA to determine which values will satisfy the equation.
x2 = y2 + 1 and y ≠ 0.
| Quantity A | Quantity B |
| x4 | y4 + 1 |
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Solution
Plugging in 2 for y gives you x2 = 5 in the given equation and 17 for Quantity B. Squaring this gives you x4 = 25 for Quantity A, which is therefore larger. Plugging in any other number gives the same result. Alternatively, doing algebra by squaring both sides of the given equation reveals Quantity A: x4 = (y2 + 1)(y2 + 1) = y4 + 2y2 + 1.The only difference between Quantities A and B is the 2y2 in Quantity A.You are told that y ≠ 0, so 2y2 is always positive, and Quantity A will always therefore be larger.The answer is choice (A).
What is the greatest value of x for which (3x – 2)(x + 1) = 0 ?
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Solution
The expression on the left side of the equation will equal zero when either (3x – 2) = 0 or (x + 1) = 0. Solving these equations yields x=2⁄3 or x = –1.The question asks you for the greatest value of x, so the answer is choice (C).
The net profit that Ann makes from selling x pillows is given by the expression x2 – 2x – 288.
| Quantity A | Quantity B |
| The number of pillows that Ann must sell for her net profit to be zero | 20 |
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Solution
Set the expression equal to zero and then factor it.You are looking for factors of 288 that have a difference of 2. So find the integer factor pairs, starting with 1: 1 and 288; 2 and 144; 3 and 96; 4 and 72; 6 and 48; 8 and 36; 9 and 32; 12 and 24; 16 and 18.The last pair you found works, so the factored form of your equation is: (x – 18)(x + 16) = 0.The solutions are 18 and –16, but obviously Ann cannot sell a negative number of pillows.The answer is choice (B).
A rectangle is formed by increasing two opposite sides of a square of side length x by y units, and decreasing the two remaining sides of the square by y units. What is the area of the rectangle?
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Solution
The dimensions of the new rectangle will be x + y and x – y.To find the area of the rectangle, multiply the length by the width: (x + y)(x – y) = x2 – y2.The answer is choice (E). Or, you can just plug in values for x and y.
If the difference between two numbers is 4, then which of the following would be sufficient to determine the value of each of the numbers?
Indicate all possible values.
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Solution
A,E, and G
Translate the question and answer choices into algebra.You are given that x – y = 4.Choice (A) tells you that x + y = 4, and you can solve these equations simultaneously by stacking them and adding to get 2x = 8, x = 4 and y = 0.Choice (A) is sufficient and correct.Choice (B) tells you that x2 – y2 = 16, and can be factored: x2 – y2 = (x + y)(x – y) = 16.You are given that (x –y) = 4, so (x + y) must also equal 4 and for that to happen, x = 4 and y = 0.Choice (B) is also sufficient and correct.Choice (C) states (x – y)2 = 16.This is simply the result of squaring what you were already given and you have no way to determine what the values of x and y are, making this choice incorrect.Choices (D) and (F) are inequalities, which means there will be multiple numbers that can work with the criteria given; eliminate both choices.Choice (E) tells you that the greater number is 4. Since x – y = 4, that now means the smaller number must be 0, making choice (E) sufficient. Finally, choice (G) states xy = 0, so at least one of the numbers must be 0. Since you were also given x – y = 4 and that neither number is negative, this means the other number must be 4.Choice (G) is sufficient and correct.
If m > 0 and 3m2 + 12m – 15 = 0, then m =
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Solution
1
Factor the quadratic equation: (3m – 3)(m + 5) = 0. Only the first factor gives a positive result: If 3m – 3 = 0, then m = 1. Alternately, you may notice that the equation is true if you ignore the variables, and making m = 1 would allow you to disregard them.
(y – 1)(y + 5) = 0
| Quantity A | Quantity B |
| y | 3 |
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Solution
If (y – 1)(y + 5) = 0, (y – 1) = 0 or (y + 5) = 0. So, y could be 1 or –5.Thus, Quantity B is greater.
a > 0
| Quantity A | Quantity B |
| (–a – 10)(10 + a) | 10 |
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Solution
FOIL out Quantity A to find –10a – a2 – 100 – 10a, or –a2 – 20a – 100. Anything other than zero to an even power is positive, so –a2 is negative. A negative number minus a positive number (20a) will remain negative. A negative minus 100 will be even more negative. So, Quantity A must be negative, and it must be less than Quantity B.The answer is choice (B). Alternatively, plugging in a few positive values for a will give you, in the parentheses: (negative) times (positive) = negative for Quantity A, except if a = 10, which yields 0 for Quantity A.But Quantity A is still less than Quantity B.