Question

The quadratic equation x^2 + 4x - 32 = 0 can be solved by factoring. Which of the following is the factored form?

Answer 1

To factor the quadratic equation \( x^2 + 4x - 32 = 0 \), we need to rewrite it in the form \( (x + a)(x + b) = 0 \) where \( a \) and \( b \) are numbers that satisfy the conditions:

1. \( a + b = 4 \) (the coefficient of the \( x \) term)
2. \( ab = -32 \) (the constant term)

Let's find two numbers that multiply to \(-32\) and add up to \(4\).

After listing the pairs of factors of \(-32\):
- \(1\) and \(-32\)
- \(-1\) and \(32\)
- \(2\) and \(-16\)
- \(-2\) and \(16\)
- \(4\) and \(-8\)
- \(-4\) and \(8\)

The pair of factors that satisfy both conditions \(a + b = 4\) and \(ab = -32\) is \(-4\) and \(8\).

Thus, we can write the factored form of the quadratic:

\[
(x - 4)(x + 8) = 0
\]

So, the factored form of the equation \( x^2 + 4x - 32 = 0 \) is:

\[
(x - 4)(x + 8) = 0
\]