When trying to solve the quadratic equation x2+4x−5=7

Question

, 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 \(x = 2\)