Esma and Hunter were trying to solve the equation:

Question

\[x^2+8x=2x-8\]
They both agreed they should first subtract
\[2x\] and add
\[8\] to both sides and rewrite the equation as
\[x^2+6x+8=0\].
Then Esma said, "I can complete the square. I can add
\[1\] to both sides to rewrite this equation as
\[x^2+6x+9=1\], factor, and solve using square roots."
Hunter said, “I'll factor the left-hand side of the equation as
\[(x+2)(x+4)\] and solve using the zero product property.”
Whose solution strategy would work?
Choose 1 answer:
Choose 1 answer:
(Choice A) Only Esma's
A
Only Esma's
(Choice B) Only Hunter's
B
Only Hunter's
(Choice C) Both
C
Both
(Choice D) Neither
D
Neither

Answer 1

Let's analyze the approaches of Esma and Hunter.

The original equation is:

\[ x^2 + 8x = 2x - 8 \]

By subtracting \(2x\) and adding \(8\) to both sides, they rewrote the equation as:

\[ x^2 + 6x + 8 = 0 \]

Esma's Approach: Esma suggests completing the square by adding \(1\) to both sides to rewrite the equation as:

\[ x^2 + 6x + 9 = 1 \]

This can be factored as:

\[ (x + 3)^2 = 1 \]

Then, by taking the square root of both sides, we obtain:

\[ x + 3 = 1 \quad \text{or} \quad x + 3 = -1 \]

This leads to:

\[ x = -2 \quad \text{or} \quad x = -4 \]

Hunter's Approach: Hunter attempts to factor the left-hand side as:

\[ (x + 2)(x + 4) = 0 \]

From this, he would arrive at:

\[ x + 2 = 0 \quad \text{or} \quad x + 4 = 0 \]

This also gives:

\[ x = -2 \quad \text{or} \quad x = -4 \]

Conclusion: Both methods are valid and lead to the same solutions. Therefore, the answer is:

(Choice C) Both