To rewrite the quadratic equation [tex]\(6(x+5)^2 + 5(x+5) - 4 = 0\)[/tex] as a simpler quadratic equation, we need to choose an appropriate substitution.
Let's examine the given equation step-by-step:
1. Identify a suitable substitution:
The expression inside the parentheses, [tex]\((x + 5)\)[/tex], is repeated in both the quadratic and linear terms of the equation. We can simplify the equation by introducing a substitution for this repeated expression. Let [tex]\( u = (x + 5) \)[/tex].
2. Apply the substitution:
Using [tex]\( u = (x + 5) \)[/tex], we rewrite the terms involving [tex]\((x + 5)\)[/tex]:
- The term [tex]\( (x + 5) \)[/tex] becomes [tex]\( u \)[/tex].
- The term [tex]\( (x + 5)^2 \)[/tex] becomes [tex]\( u^2 \)[/tex].
3. Rewrite the entire equation:
Substitute [tex]\( u \)[/tex] into the original equation:
[tex]\[
6(x + 5)^2 + 5(x + 5) - 4 = 0
\][/tex]
transforms into:
[tex]\[
6u^2 + 5u - 4 = 0
\][/tex]
By making the substitution [tex]\( u = (x + 5) \)[/tex], we transform the original equation into a simpler quadratic form.
So the correct substitution to rewrite [tex]\( 6(x + 5)^2 + 5(x + 5) - 4 = 0 \)[/tex] as a quadratic equation is:
[tex]\( u = (x + 5) \)[/tex].
Therefore, the correct answer is:
[tex]\[ u = (x + 5) \][/tex]
