To find the factored form of the given quadratic equation [tex]\(6x^2 + 2x - 8 = 0\)[/tex], we follow several steps:
1. Identify the quadratic equation:
The given quadratic equation is [tex]\(6x^2 + 2x - 8 = 0\)[/tex].
2. Factor out the greatest common factor:
Notice that in the equation [tex]\(6x^2 + 2x - 8 = 0\)[/tex], each term can be factored by the greatest common factor (GCF), which is 2.
[tex]\[
6x^2 + 2x - 8 = 2(3x^2 + x - 4)
\][/tex]
3. Factor the quadratic expression [tex]\(3x^2 + x - 4\)[/tex]:
We need to factor [tex]\(3x^2 + x - 4\)[/tex] into two binomials. This can be done by finding two numbers that multiply to [tex]\((3 \cdot -4)\)[/tex] and add up to [tex]\(1\)[/tex].
Here, we are looking for numbers that multiply to [tex]\(-12\)[/tex] (since [tex]\(3 \times -4 = -12\)[/tex]) and add up to [tex]\(1\)[/tex].
The two numbers that satisfy this are [tex]\(4\)[/tex] and [tex]\(-3\)[/tex] because:
[tex]\[
4 \times (-3) = -12 \quad \text{and} \quad 4 + (-3) = 1.
\][/tex]
4. Rewriting the middle term using these factors:
Rewrite [tex]\(3x^2 + x - 4\)[/tex] by splitting the middle term using [tex]\(4\)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[
3x^2 + 4x - 3x - 4
\][/tex]
5. Factor by grouping:
Group the terms to factor them separately:
[tex]\[
(3x^2 + 4x) - (3x + 4)
\][/tex]
From the first group, factor out [tex]\(x\)[/tex]:
[tex]\[
x(3x + 4) - 1(3x + 4)
\][/tex]
6. Factor out the common binomial factor:
[tex]\[
(x - 1)(3x + 4)
\][/tex]
Thus, the factored form of [tex]\(3x^2 + x - 4\)[/tex] is [tex]\((x - 1)(3x + 4)\)[/tex].
7. Incorporate the GCF factor:
[tex]\[
6x^2 + 2x - 8 = 2(3x^2 + x - 4) = 2(x - 1)(3x + 4)
\][/tex]
Therefore, the correctly factored form of the given equation is:
[tex]\[
2(3x + 4)(x - 1) = 0
\][/tex]
From the given choices,
- [tex]\(2(3x + 4)(x + 1) = 0\)[/tex]
- [tex]\(2(3x + 4)(x - 1) = 0\)[/tex]
- [tex]\((6x + 1)(x - 8) = 0\)[/tex]
The correct answer is:
[tex]\[
\boxed{2(3x + 4)(x - 1) = 0}
\][/tex]
