High School

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What substitution should be used to rewrite [tex]$4x^4 - 21x^2 + 20 = 0$[/tex] as a quadratic equation?

A. [tex]u = x^2[/tex]

B. [tex]u = 2x^2[/tex]

C. [tex]u = x^4[/tex]

D. [tex]u = 4x^4[/tex]

Answer :

To rewrite the equation [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex] as a quadratic equation, we need to use a substitution method.

Let's follow these steps:

1. Identify a substitution that simplifies the given equation into a quadratic form. In this case, we will substitute [tex]\( u = x^2 \)[/tex].

2. Substitute [tex]\( u = x^2 \)[/tex] into the original equation. Notice that [tex]\( x^4 \)[/tex] can be written as [tex]\( (x^2)^2 \)[/tex], which is [tex]\( u^2 \)[/tex].

3. Rewrite the original equation [tex]\(4x^4 - 21x^2 + 20\)[/tex] using the substitution [tex]\( u = x^2 \)[/tex]:

[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]

4. Simplify this expression:

[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]

Now the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is a quadratic equation in terms of [tex]\( u \)[/tex].

Hence, the correct substitution to rewrite [tex]\( 4x^4 - 21x^2 + 20 = 0 \)[/tex] as a quadratic equation is [tex]\( u = x^2 \)[/tex].

So, the answer is:
[tex]\[
u = x^2
\][/tex]

This corresponds to the first choice in your options.

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