Which quadratic equation is equivalent to [tex]\left(x^2-1\right)^2-11\left(x^2-1\right)+24=0[/tex]?

A. [tex]u^2-11u+24=0[/tex] where [tex]u=\left(x^2-1\right)[/tex]
B. [tex](u^2)^2-11(u^2)+24=0[/tex] where [tex]u=\left(x^2-1\right)[/tex]
C. [tex]u^2+1-11u+24=0[/tex] where [tex]u=\left(x^2-1\right)[/tex]
D. [tex](u^2-1)^2-11(u^2-1)+24=0[/tex] where [tex]u=\left(x^2-1\right)[/tex]



Answer :

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]