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rss GMAT Practice Question (One of Hundreds)
f(x) = x2 + 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
If f(-6) = 0, (x + 6) is a factor. Reverse factor the equation with this fact in mind.
  1. 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.
  2. Method 1: Solve for K.
    1. f(x) = x2 + 4x + k = 12
      f(x) = x2 + 4x + k - 12 = 0 (subtract 12)
    2. 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
    3. The equation is now:
      f(x) = x2 + 4x + 0 - 12 = 0
      f(x) = x2 + 4x - 12 = 0
    4. By factoring: x2 + 4x - 12 = 0 equals:
      (x+6)(x+n) Note: x+6 is from f(-6)=0
    5. Since we need two numbers that add to +4 and multiply to -12, we know that +6 and -2 work. Consequently, (x-2) is a factor and therefore, f(+2)=0, so n = 2.
  3. Method 2: Ignore K.
    1. A crucial insight in unlocking this problem is recognizing that if f(a) = 0 and f(x) is a quadratic equation in the form x2 + bx + c = 0, (x – a) is a factor of the equation. In order to get the equation into quadratic form, subtract 12.
      f(x) = x2 + 4x + k = 12
      f(x) = x2 + 4x + k – 12 = 0
    2. Since f(-6) = 0, (x+6) is a factor.
    3. It is important to remember how factoring works. Specifically, remember the following:
      (x + d)(x + e) = x2 + dex + de = 0
      So: (x + 6)(x + a) = x2 + 4x + (k – 12)
    4. 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.
  4. Answer B is correct.

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