Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x^{2} + 4x - 32 = (x - 4)(x + 8) \)
Solution
To factor the quadratic expression \( x^{2} + 4x - 32 \), we'll look for two numbers that multiply to \(-32\) (the constant term) and add up to \(4\) (the coefficient of the middle term).
**Step 1: Identify the Pair of Numbers**
We need two numbers, let's call them \(m\) and \(n\), such that:
- \(m \times n = -32\)
- \(m + n = 4\)
Let's list the possible pairs of factors of \(-32\):
- \(1\) and \(-32\) (sum = \(-31\))
- \(-1\) and \(32\) (sum = \(31\))
- \(2\) and \(-16\) (sum = \(-14\))
- \(-2\) and \(16\) (sum = \(14\))
- \(4\) and \(-8\) (sum = \(-4\))
- \(-4\) and \(8\) (sum = \(4\))
The pair \(-4\) and \(8\) adds up to \(4\), which is what we need.
**Step 2: Write the Factored Form**
Using these numbers, the factored form of the quadratic is:
\[
x^{2} + 4x - 32 = (x - 4)(x + 8)
\]
**Verification:**
\[
(x - 4)(x + 8) = x^2 + 8x - 4x - 32 = x^2 + 4x - 32
\]
This confirms that the factorization is correct.
**Final Answer:**
\[
x^{2} + 4x - 32 = (x - 4)(x + 8)
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Extra Insights
To factor the quadratic expression \( x^{2} + 4x - 32 \), we want to find two numbers that multiply to \(-32\) (the constant term) and add up to \(4\) (the coefficient of the linear term). The numbers \(8\) and \(-4\) fit the bill since \(8 \times -4 = -32\) and \(8 + (-4) = 4\). Therefore, we can rewrite the expression as \( (x + 8)(x - 4) \). Now you can check your work by expanding the factored form! Multiply \( (x + 8)(x - 4) \) back to see if you return to the original quadratic equation. It’s a great way to confirm your factoring skills!
