Certainly! Let's go step-by-step to rewrite the given quadratic equation using substitution.
We start with the given equation:
[tex]\[
(3x + 2)^2 + 7(3x + 2) - 8 = 0
\][/tex]
1. Substitute [tex]\( u = 3x + 2 \)[/tex]:
This substitution helps to simplify the equation. Thus, we replace every instance of [tex]\( 3x + 2 \)[/tex] with [tex]\( u \)[/tex].
[tex]\[
u^2 + 7u - 8 = 0
\][/tex]
2. Rearrange the equation:
At this stage, the equation is now a standard quadratic equation in terms of [tex]\( u \)[/tex].
[tex]\[
u^2 + 7u - 8 = 0
\][/tex]
3. Solve the quadratic equation for [tex]\( u \)[/tex]:
Now, we need to find the values of [tex]\( u \)[/tex] that satisfy the equation [tex]\( u^2 + 7u - 8 = 0 \)[/tex]. This can be done using the quadratic formula:
[tex]\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -8 \)[/tex]. Plugging these into the formula, we get:
[tex]\[
u = \frac{-7 \pm \sqrt{7^2 - 4 \times 1 \times (-8)}}{2 \times 1}
\][/tex]
[tex]\[
u = \frac{-7 \pm \sqrt{49 + 32}}{2}
\][/tex]
[tex]\[
u = \frac{-7 \pm \sqrt{81}}{2}
\][/tex]
[tex]\[
u = \frac{-7 \pm 9}{2}
\][/tex]
Thus, we obtain two solutions for [tex]\( u \)[/tex]:
[tex]\[
u = \frac{-7 + 9}{2} = 1 \quad \text{and} \quad u = \frac{-7 - 9}{2} = -8
\][/tex]
4. Back Substitution:
Having found the values of [tex]\( u \)[/tex], we now substitute back [tex]\( u = 3x + 2 \)[/tex].
For [tex]\( u = 1 \)[/tex]:
[tex]\[
1 = 3x + 2 \Rightarrow 3x = 1 - 2 \Rightarrow 3x = -1 \Rightarrow x = -\frac{1}{3}
\][/tex]
For [tex]\( u = -8 \)[/tex]:
[tex]\[
-8 = 3x + 2 \Rightarrow 3x = -8 - 2 \Rightarrow 3x = -10 \Rightarrow x = -\frac{10}{3}
\][/tex]
Thus, the required solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = -\frac{1}{3} \quad \text{and} \quad x = -\frac{10}{3}
\][/tex]
The step-by-step process ensures that we've accurately solved for [tex]\( x \)[/tex] by utilizing substitution and solving the quadratic equation correctly.
