To determine which quadratic equation is equivalent to [tex]\(\left(x^2 - 1\right)^2 - 11\left(x^2 - 1\right) + 24 = 0\)[/tex], follow these steps:
1. Define the substitution:
Let [tex]\( u = x^2 - 1 \)[/tex].
2. Substitute:
Replace every occurrence of [tex]\( x^2 - 1 \)[/tex] in the original equation with [tex]\( u \)[/tex]. The original equation becomes:
[tex]\[
(u)^2 - 11(u) + 24 = 0.
\][/tex]
3. Simplify:
Simplify the resulting quadratic equation:
[tex]\[
u^2 - 11u + 24 = 0.
\][/tex]
4. Compare with the given choices:
- The first choice is : [tex]\( u^2 - 11u + 24 = 0 \)[/tex], which matches our simplified equation.
- The second choice is: [tex]\( (u^2)^2 - 11(u^2) + 24 \)[/tex]. This is not equivalent to our simplified equation.
- The third choice is: [tex]\( u^2 + 1 - 11u + 24 = 0 \)[/tex]. This is also not equivalent to our simplified equation.
- The fourth choice is: [tex]\( (u^2 - 1)^2 - 11(u^2 - 1) + 24 \)[/tex]. This is not equivalent to our simplified equation either.
Thus, the correct equivalent quadratic equation is the first choice:
[tex]\[
u^2 - 11u + 24 = 0,
\][/tex]
where [tex]\( u = x^2 - 1 \)[/tex].
