To factor the quadratic equation \(x^2 + 4x - 32 = 0\) by grouping, we first need to express the quadratic in a form that allows us to factor by finding two numbers that multiply to \(-32\) (the constant term) and add to \(4\) (the coefficient of the linear term).
-
We look for two numbers that multiply to \(-32\) and add to \(4\). The pairs of factors of \(-32\) are:
- \(1\) and \(-32\)
- \(-1\) and \(32\)
- \(2\) and \(-16\)
- \(-2\) and \(16\)
- \(4\) and \(-8\)
- \(-4\) and \(8\)
The correct pair is \(8\) and \(-4\), since: \[ 8 + (-4) = 4 \quad \text{and} \quad 8 \times (-4) = -32 \]
-
We can now rewrite the middle term \(4x\) using \(8x\) and \(-4x\): \[ x^2 + 8x - 4x - 32 = 0 \]
-
Next, we group the terms: \[ (x^2 + 8x) + (-4x - 32) = 0 \]
-
Now we factor out the greatest common factor from each group: \[ x(x + 8) - 4(x + 8) = 0 \]
-
We can factor out the common binomial factor \((x + 8)\): \[ (x - 4)(x + 8) = 0 \]
Thus, the factored form of the quadratic equation \(x^2 + 4x - 32 = 0\) is: \[ (x - 4)(x + 8) = 0 \]