Question

Martin solved the quadratic equation 3x^2+2x−8=0
. His work is shown below. However, he accidentally spilled ink over the step where he divided by the GCF that he found for each pair. Which of the following options shows the missing part of Martin’s work?

Several lines of equations.

(1 point)
Responses

3x(x+2)−4(x−2)=0
3 x left parenthesis x plus 2 right parenthesis minus 4 left parenthesis x minus 2 right parenthesis equals 0

3x(x+2)+4(x+2)=0
3 x left parenthesis x plus 2 right parenthesis plus 4 left parenthesis x plus 2 right parenthesis equals 0

3x(x+2)−4(x+2)=0
3 x left parenthesis x plus 2 right parenthesis minus 4 left parenthesis x plus 2 right parenthesis equals 0

x(3x+6)+4(x+2)=0

Answer 1

To solve the quadratic equation \(3x^2 + 2x - 8 = 0\) and find the missing part of Martin's work, we look for the factorization of the equation.

The equation can be rewritten by grouping terms. The goal is to factor the quadratic, and we can do this by identifying a common factor from pairs of terms.

Let's analyze Martin's options:

1. \(3x(x+2)−4(x−2)=0\)
2. \(3x(x+2)+4(x+2)=0\)
3. \(3x(x+2)−4(x+2)=0\)
4. \(x(3x+6)+4(x+2)=0\)

To find the correct missing part, we should first identify the greatest common factor (GCF) for the coefficients in the quadratic expression.

1. Start by looking at the polynomial \(3x^2 + 2x - 8\).
2. Try re-grouping or factoring:
- We need two numbers that multiply to \(3 \cdot (-8) = -24\) and add to \(2\). The numbers \(6\) and \(-4\) work here.

This gives us:

\[
3x^2 + 6x - 4x - 8 = 0
\]

Now, group the terms:

\[
(3x^2 + 6x) + (-4x - 8) = 0
\]

Factor out the GCF from each group:

\[
3x(x + 2) - 4(x + 2) = 0
\]

Now, we see both groups have a common factor of \((x + 2)\):

\[
(3x - 4)(x + 2) = 0
\]

This means the correct missing part that fits into his original work is:

\[
3x(x+2)−4(x+2)=0
\]

So the response that shows the missing part of Martin’s work is:

**3 x left parenthesis x plus 2 right parenthesis minus 4 left parenthesis x plus 2 right parenthesis equals 0**.