Martin solved the quadratic equation 3x2+2x−8=0

Question

. 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 plus 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 minus 2 right parenthesis equals 0

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

Answer 1

To solve the quadratic equation \(3x^2 + 2x - 8 = 0\), we can factor it if we can express it correctly. Martin likely started by grouping the terms and factoring by grouping or finding a common factor.

Given the equation, we can first rewrite it for easier factoring:

1. The expression \(3x^2 + 2x - 8\) can be grouped as follows:

   • We notice that we can express \(2x\) as \(3x + (-x)\) which allows us to regroup.
   • Let's express it as \(3x^2 + 3x - x - 8\).

   This can be grouped into: \( (3x^2 + 3x) + (-x - 8) = 0 \)

2. Now we can factor by grouping:

   • From the first group, we can take out \(3x\), yielding \(3x(x + 1)\).
   • From the second group, we can take out \(-1\), yielding \(-1(x + 8)\).

Thus, we can rewrite the equation as: \[ 3x^2 + 3x - x - 8 = 0 \] which factors to look like this: \[ 3x(x + 1) - 1(x + 8) = 0 \]

3. However, Martin's missing step indicates a GCF was factored out, so he might have identified \(x + 2\) as common from a term expressing that quadratic differently. This appears to reflect an alternative grouping.

The most reasonable factorization performed here is to try and find the GCF of pairs leading towards zero: When Martin factored it he possibly arrived at: \[ 3x(x + 2) - 4(x + 2) = 0 \]

Thus when dividing by the common factor \(x + 2\), the step that was supposed to follow is essentially: \[ (3x - 4)(x + 2) = 0 \]

So the missing option in his work is the one that matches this factoring based on GCF, which reads: 3x\( (x + 2) \) - 4\( (x + 2) \) = 0

Hence, the correct choice is: 3x(x+2)−4(x+2)=0