To determine which quadratic equation is equivalent to [tex]\((x^2 - 1)^2 - 11(x^2 - 1) + 24 = 0\)[/tex], let's perform a substitution step-by-step.
1. Identify a substitution variable [tex]\(u\)[/tex]:
Let [tex]\(u\)[/tex] be a substitution for [tex]\(x^2 - 1\)[/tex]. This gives us:
[tex]\[
u = x^2 - 1
\][/tex]
2. Substitute [tex]\(u\)[/tex] into the original equation:
Replace every occurrence of [tex]\(x^2 - 1\)[/tex] in the equation with [tex]\(u\)[/tex]:
[tex]\[
(x^2 - 1)^2 - 11(x^2 - 1) + 24 = 0
\][/tex]
Substituting [tex]\(x^2 - 1 = u\)[/tex]:
[tex]\[
u^2 - 11u + 24 = 0
\][/tex]
3. Compare the obtained equation to the given options:
From the substitution, the equation simplifies to:
[tex]\[
u^2 - 11u + 24 = 0
\][/tex]
where [tex]\(u = x^2 - 1\)[/tex].
Now, let's see which option matches this derived equation:
- Option 1: [tex]\(u^2 - 11u + 24 = 0\)[/tex] where [tex]\(u = x^2 - 1\)[/tex]
- Option 2: [tex]\((u^2)^2 - 11(u^2) + 24\)[/tex] where [tex]\(u = x^2 - 1\)[/tex]
- Option 3: [tex]\(u^2 + 1 - 11u + 24 = 0\)[/tex] where [tex]\(u = x^2 - 1\)[/tex]
- Option 4: [tex]\((u^2 - 1)^2 - 11(u^2 - 1) + 24\)[/tex] where [tex]\(u = x^2 - 1\)[/tex]
Clearly, the first option matches our derived equation. Therefore, the correct quadratic equation equivalent to [tex]\((x^2 - 1)^2 - 11(x^2 - 1) + 24 = 0\)[/tex] is:
[tex]\[
u^2 - 11u + 24 = 0 \; \text{where} \; u = x^2 - 1
\][/tex]
