Question

When trying to solve the quadratic equation x2+4x−5=7
, George provided his work in the table.
Step 1 Begin with the equation. x^2 + 4x - 5 = 7
Step 2 Write quadratic equation as the product of two factors. (x + 5) (x - 1) = 7
Step 3 Set each factor equal to 0. x + 5 = 0 and x - 1 = 0
Step 4 Solve each equation. x = -5 and x = 1
What did he do wrong?

(1 point)
What are the correct solutions to the original equation?(2 points)
List the smaller solution first.

x=

and

Answer 1

George made a mistake in Step 2 when he incorrectly wrote the factors of the quadratic equation. Let's properly solve the equation step by step.

1. **Start with the original equation:**
\[
x^2 + 4x - 5 = 7
\]

2. **Rearrange the equation to set it to 0:**
\[
x^2 + 4x - 5 - 7 = 0 \implies x^2 + 4x - 12 = 0
\]

3. **Now factor the quadratic equation \(x^2 + 4x - 12\):**
To find the factors, we need two numbers that multiply to -12 and add to 4. These numbers are 6 and -2. So, we can factor it as:
\[
(x + 6)(x - 2) = 0
\]

4. **Set each factor equal to 0:**
\[
x + 6 = 0 \quad \text{or} \quad x - 2 = 0
\]

5. **Solve each equation:**
\[
x = -6 \quad \text{and} \quad x = 2
\]

**What did George do wrong?**
- George incorrectly factored the quadratic equation as \((x + 5)(x - 1)\) instead of the correct factors \((x + 6)(x - 2)\).

**What are the correct solutions to the original equation?**
The correct solutions are:
\[
x = -6 \quad \text{and} \quad x = 2
\]

**List the smaller solution first:**
- \(x = -6\) and \(x = 2\)