To determine which quadratic equation is equivalent to [tex]\(\left(x^2-1\right)^2-11\left(x^2-1\right)+24=0\)[/tex], we will use a substitution method.
1. Start with the given equation:
[tex]\[
\left(x^2-1\right)^2-11\left(x^2-1\right)+24=0
\][/tex]
2. Introduce a substitution where we let [tex]\( u = x^2 - 1 \)[/tex].
3. Substitute [tex]\( u \)[/tex] into the original equation:
[tex]\[
(u)^2 - 11(u) + 24 = 0
\][/tex]
Hence, substituting [tex]\( u = x^2 - 1 \)[/tex] transforms the original equation into:
[tex]\[
u^2 - 11u + 24 = 0
\][/tex]
Therefore, the equivalent quadratic equation is:
[tex]\[
u^2 - 11u + 24 = 0
\][/tex]
where [tex]\( u = x^2 - 1 \)[/tex].
The correct choice from the given options is:
[tex]\[ u^2 - 11u + 24 = 0 \text{ where } u = x^2 - 1 \][/tex]
