Question

Solve using zero product property

X^2 +2x -48 =0

Answer 1

To solve the quadratic equation \( x^2 + 2x - 48 = 0 \) using the zero product property, we first need to factor the equation.

1. Look for two numbers that multiply to \(-48\) (the constant term) and add up to \(2\) (the coefficient of \(x\)).

The numbers that fit these criteria are \(8\) and \(-6\) because:
\[
8 \times (-6) = -48
\]
\[
8 + (-6) = 2
\]

2. We can factor the equation as follows:
\[
(x + 8)(x - 6) = 0
\]

3. Now, apply the zero product property, which states that if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\):
\[
x + 8 = 0 \quad \text{or} \quad x - 6 = 0
\]

4. Solve each equation:
- For \(x + 8 = 0\):
\[
x = -8
\]
- For \(x - 6 = 0\):
\[
x = 6
\]

Thus, the solutions to the equation \(x^2 + 2x - 48 = 0\) are \(x = -8\) and \(x = 6\).