McGraw-Hill Education GRE 2019

Exercise Answers

Discrete Quantitative Questions

1. D Simplify the equation by dividing both sides by 2: x^2 + 10x = −24. Since you
   have a quadratic, you should set it equal to zero: x^2 + 10x + 24 = 0. Now factor
   the quadratic: (x + 6)(x + 4) = 0. Either factor can equal zero, so:
   (x + 6) = 0
   x = −6 or (x + 4) = 0
   x = −4


2. D Notice that a^2 − 2ab + b^2 is one of the special products. The expression
   factors to (a − b)^2. If (a − b) = 4, then (a − b)^2 = 16.


3. B Note that the given equation is quadratic. There will thus be two solutions
   for (x + 3): the positive square root of 81 and the negative square root of 81.
   Thus (x + 3) = 9 → x = 6 or (x + 3) = −9 → x = −12.


4. 72 Since x^2 − y^2 is a difference of squares, you can factor the original equation
   to (x + y)(x − y) = 12. Substitute 4 for (x + y) in the original equation:
   4(x − y) = 12. Divide both sides of the equation by 4: (x − y) = 3. So you know
   that x − y = 3 and that x + y = 4. To solve for x, you will add the equations:
   x − y = 3
   x + y = 4
   2 x = 7
   ↓
   x =^72


5. D Expand the expression on the left side of the first equation: a^2 + 2ab + b^2 = 36.
   Substitute 4 for ab: a^2 + 8 + b^2 = 36. Subtract 8 from both sides to solve for (a^2
   
   
   • b^2 ): a^2 + b^2 = 28.
   
   
   
   


6. B To isolate the variables, multiply both sides of the equation by (2x −2y):
   (2x + 2y)(2x − 2y) = 1. Notice that (2x + 2y)(2x − 2y) is in the form of (a + b)(a
   − b). Since (a + b)(a − b) becomes a^2 − b^2 after applying FOIL, (2x + 2y)
   (2x − 2y) becomes (2x)^2 − (2y)^2 = 4x^2 − 4y^2 after applying FOIL. The equation
   is now 4x^2 − 4y^2 = 1. Factor 4 from both terms on the left side: 4(x^2 − y^2 ) = 1.
   Divide by 4: x^2 − y^2 =^14.


7. − 180 Notice that the right side of the equation is the factored form of the left
   side. How do you factor a common quadratic? Think of a simpler situation: x^2
   
   
   • 5x + 6 = (x + 3)(x +2). Why? Because 3 and 2 multiply to yield 6 and add to
     yield 5. So in the original equation, −10 and 18 multiply to yield k. k = −18 0.
   
   
   
   


8. D Notice that the right side of the equation is the factored form of the left side.
   How do you factor a common quadratic? Think of a simpler situation: x^2 + 5x
   
   
   • 6 = (x + 3)(x +2). Why? Because 3 and 2 multiply to yield 6 and add to yield
   
   
   
   
   
   
   5. So in the original equation, −z and q multiply to yield b. Thus b = −qz.
   
   
   
   


9. A Note that the expression on the left side of the equation is the factored form
   of a difference of squares: (x + y)(x − y) = x^2 − y^2 , so (√a − √b)(√a + √b) =
   √a

2

− √b

2

= a − b. Thus a − b = 12. Isolate a: a = b + 12.

290 PART 4 ■ MATH REVIEW

03-GRE-Test-2018_173-312.indd 290 12/05/17 11:55 am