Question

Use the tables to answer the question.

Jamal
This table displays the steps followed by Jamal to solve a given quadratic equation.
Step 1: Set equation =0
. x2+4x−12=0
Step 2: Write quadratic equation as the product of two factors. (x+6)(x−2)=0
Step 3: Set each factor equal to 0. x+6=0
and x−2=0
Step 4: Solve each equation. x=−6
and x=2
George
This table outlines the steps to solve the quadratic equation by factoring, leading to the solutions x=−6 and x=2.
Step 1: Begin with the equation. x2+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
When trying to solve the quadratic equation x2+4x−5=7
, Jamal and George each provided their work in the tables. Each said they used the Zero Product Property to solve the equation after step 2. Explain how you know the correct student’s solution is accurate and how you know the incorrect student’s solution is inaccurate.

Answer 1

Jamal is correct; George is not.

Reasoning:
- Start with the given equation x^2 + 4x − 5 = 7. Subtract 7 from both sides to set the equation to 0: x^2 + 4x − 12 = 0. (This is what Jamal did.)
- Factor: x^2 + 4x − 12 = (x + 6)(x − 2) = 0. By the Zero Product Property, x + 6 = 0 or x − 2 = 0, so x = −6 or x = 2.
- Check by substitution:
- x = −6: (−6)^2 + 4(−6) − 5 = 36 − 24 − 5 = 7 ✓
- x = 2: 2^2 + 4(2) − 5 = 4 + 8 − 5 = 7 ✓

Why George is wrong:
- George correctly factored the left side as (x + 5)(x − 1), but he left the equation as (x + 5)(x − 1) = 7. The Zero Product Property only applies when a product equals 0, not 7. You cannot conclude x + 5 = 0 or x − 1 = 0 from (x + 5)(x − 1) = 7.
- If you test George’s proposed roots x = −5 or x = 1 in the original equation, each gives 0 = 7, which is false. Hence George’s application of the Zero Product Property is invalid.