Certainly! Let's work through the given question step-by-step and find the equivalent quadratic equation using substitution.
We start with the given equation:
[tex]\[
(3x + 2)^2 + 7(3x + 2) - 8 = 0
\][/tex]
To simplify this equation, we use a substitution method. Let's set:
[tex]\[
u = 3x + 2
\][/tex]
Now, substitute [tex]\(u\)[/tex] into the given equation:
[tex]\[
(u)^2 + 7u - 8 = 0
\][/tex]
This substitution transforms our original equation into a simpler quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[
u^2 + 7u - 8 = 0
\][/tex]
Now we need to clarify the correct substitution from the given choices. The choices are:
1. [tex]\(u^2 + 7u - 8 = 0, \text{ where } u = (3x + 2)^2\)[/tex]
2. [tex]\(u^2 + 7u - 8 = 0, \text{ where } u = 3x + 2\)[/tex]
3. [tex]\(u^2 + 7u - 8 = 0, \text{ where } u = 7(3x + 2)\)[/tex]
4. [tex]\(u^2 + u - 8 = 0\)[/tex]
From our substitution step, we know:
[tex]\[
u = 3x + 2
\][/tex]
Thus, the correct substitution and the equivalent quadratic equation are:
[tex]\[
u^2 + 7u - 8 = 0, \text{ where } u = 3x + 2
\][/tex]
This explains how we arrived at the equivalent quadratic equation by using the substitution [tex]\(u = 3x + 2\)[/tex].
