S99-07

Official Solution:

Given that \(x^4 - 25x^2 = -144\), which of the following is NOT a sum of two possible values of \(x\)?

A. -7
B. -1
C. 0
D. 3
E. 7


First, solve the given equation:
\(x^4 - 25x^2 = -144\)
\(x^4 - 25x^2 + 144 = 0\)

Substitute \(z = x^2\):
\(z^2 - 25z + 144 = 0\)

Use trial and error to find a pair of factors of 144 that sum to 25. Start with 12 & 12, which sum to 24; then try 16 and 9, which work.

\((z - 16)(z - 9) = 0\)

\(z = 16\) or \(z = 9\)

Now put \(x^2\) back in:

\(x^2 = 16\) or \(x^2 = 9\)

\(x = -4\) or \(4\), or \(x = -3\) or \(3\). There are four possible solutions.

Possible sums of two solutions include the following:
\(-7 = -4 + -3\)
\(-1 = -4 + 3\)

\(0 = -4 + 4\) or \(-3 + 3\)
\(7 = 4 + 3\)

However, 3 itself is NOT a possible sum of two solutions of the equation.


Answer: D