Practice GMAT Problem Solving Question
Return to the list of practice GMAT problem solving questions.
f(x) = x^{2} + 4x + k = 12; f(6) = 0; if k is a constant and n is the number for which f(n) = 0, what is the value of n?
Correct Answer: B
 There are two main approaches you can take to solving this problem. You can either solve for k and turn this problem into a simple factoring problem or you can ignore k and reverse factor.

Method 1: Solve for K.
 f(x) = x^{2} + 4x + k = 12
f(x) = x^{2} + 4x + k  12 = 0 (subtract 12)  Plug in x=6 knowing that the value of f(x) must equal zero since f(6)=0.
f(6) = (6)^{2} + 4(6) + k  12 = 0
36 24 + k  12 = 0
0  k = 0 > k = 0  The equation is now:
f(x) = x^{2} + 4x + 0  12 = 0
f(x) = x^{2} + 4x  12 = 0  By factoring: x^{2} + 4x  12 = 0 equals:
(x+6)(x+n) Note: x+6 is from f(6)=0  Since we need two numbers that add to +4 and multiply to 12, we know that +6 and 2 work. Consequently, (x2) is a factor and therefore, f(+2)=0, so n = 2.
 f(x) = x^{2} + 4x + k = 12

Method 2: Ignore K.
 A crucial insight in unlocking this problem is recognizing that if f(a) = 0 and f(x) is a quadratic equation in the form x^{2} + bx + c = 0, (x – a) is a factor of the equation. In order to get the equation into quadratic form, subtract 12.
f(x) = x^{2} + 4x + k = 12
f(x) = x^{2} + 4x + k – 12 = 0  Since f(6) = 0, (x+6) is a factor.
 It is important to remember how factoring works. Specifically, remember the following:
(x + d)(x + e) = x^{2} + dex + de = 0
So: (x + 6)(x + a) = x^{2} + 4x + (k – 12)  With this in mind, you know that 6 + a = 4. So, a = 2 and therefore, (x  2) is the other factor of the quadratic. So, f(2) = 0 and n = 2.
 A crucial insight in unlocking this problem is recognizing that if f(a) = 0 and f(x) is a quadratic equation in the form x^{2} + bx + c = 0, (x – a) is a factor of the equation. In order to get the equation into quadratic form, subtract 12.
 Answer B is correct.
Return to the list of practice GMAT problem solving questions.