Use the tables to answer the question.

Question

Jamal
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
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.

(2 points)

Answer 1

To determine the accuracy of Jamal's and George's solutions, we need to carefully analyze their steps, particularly focusing on the initial equations and how they proceeded to factor them.

Step-by-step Analysis:

1. Jamal's Work:

   • Step 1: He sets the equation \( x^2 + 4x - 12 = 0 \). This is incorrect because the original equation given was \( x^2 + 4x - 5 = 7 \). The correct first step should be to rearrange it to \( x^2 + 4x - 12 = 0 \) by moving 7 to the left side subtractively.
   • Step 2: Jamal factors it as \( (x + 6)(x - 2) = 0 \). However, this factorization is not correct for the actual equation \( x^2 + 4x - 12 \).

2. George's Work:

   • Step 1: He starts with the correct equation \( x^2 + 4x - 5 = 7 \) but fails to rearrange it correctly as \( x^2 + 4x - 12 = 0 \). Instead, he sets up \( x^2 + 4x - 5 - 7 = 0 \) directly.
   • Step 2: George writes \( (x + 5)(x - 1) = 7 \), which is also incorrect. The correct factoring should yield \( (x + 2)(x - 2) = 12 \), so neither student gets the correct factors.

Conclusions:

• Jamal's Solution Inaccuracy: Jamal's first step is entirely wrong, as he sets his quadratic equation incorrectly. This leads to incorrect factors, and thus his application of the Zero Product Property fails because the factors he provides do not reflect the original quadratic equation derived from the problem. Therefore, the roots he calculates, \( x = -6 \) and \( x = 2 \), are invalid concerning the original equation.

• George's Solution Inaccuracy: George found the equation correctly but made an error at the next step of factoring (mixing the numbers leading to \( 7 \)). His evaluation misses the rearrangement into the form from which appropriate factoring can occur. Although he correctly attempted to apply the Zero Product Property, it was irrelevant because his factors and products did not stem from the correct equation.

In summary: Both students made critical errors in their equations. While Jamal's factorization stems from an incorrect starting point, George began accurately but also inaccurately factored it. Therefore, neither correctly applied the Zero Product Property as their factors were not derived from the same accurate quadratic expression.