To rewrite the given equation [tex]\( 6(x+5)^2 + 5(x+5) - 4 = 0 \)[/tex] as a quadratic equation, we can use a substitution that simplifies the expression. The goal is to transform [tex]\( (x+5) \)[/tex] into a simpler variable. Let's follow the steps:
1. Identify the expression to simplify: [tex]\( (x+5) \)[/tex] appears twice in the equation.
2. Choose an appropriate substitution:
Set [tex]\( u = (x + 5) \)[/tex].
With this substitution, the given equation [tex]\( 6(x+5)^2 + 5(x+5) - 4 = 0 \)[/tex] can now be rewritten in terms of [tex]\( u \)[/tex]:
3. Substitute [tex]\( u \)[/tex] into the equation:
- Wherever [tex]\( (x+5) \)[/tex] appears, replace it with [tex]\( u \)[/tex]:
[tex]\[
6u^2 + 5u - 4 = 0
\][/tex]
Thus, the equation [tex]\( 6(x+5)^2 + 5(x+5) - 4 = 0 \)[/tex] transforms into the quadratic equation [tex]\( 6u^2 + 5u - 4 = 0 \)[/tex] using the substitution [tex]\( u = (x + 5) \)[/tex].
Therefore, the correct substitution to rewrite the given equation as a quadratic equation is:
[tex]\[ u = (x + 5) \][/tex]
