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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]\left(u^2\right)^2-11\left(u^2\right)+24[/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]\left(u^2-1\right)^2-11\left(u^2-1\right)+24[/tex] where [tex]u=\left(x^2-1\right)[/tex]


Sagot :

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]