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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, you can use substitution. The goal is to make the equation resemble the standard form of a quadratic, which is [tex]\(au^2 + bu + c = 0\)[/tex].

Here's a step-by-step explanation:

1. Identify the Substitution:
- Look at the terms in the equation: [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].
- Notice that [tex]\(x^4\)[/tex] is the square of [tex]\(x^2\)[/tex] (i.e., [tex]\((x^2)^2 = x^4\)[/tex]).
- This suggests using the substitution [tex]\(u = x^2\)[/tex].

2. Apply the Substitution:
- Substitute [tex]\(u = x^2\)[/tex] into the equation.
- The original equation becomes:
[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]
- Simplify the expression:
[tex]\[
4u^2 - 21u + 20 = 0
\][/tex]

3. Interpret the Result:
- Now, the equation [tex]\(4u^2 - 21u + 20 = 0\)[/tex] is in the standard quadratic form, where [tex]\(u\)[/tex] is the variable.

Therefore, the substitution needed to rewrite the original equation as a quadratic is [tex]\(u = x^2\)[/tex]. This transforms the equation into a standard quadratic form.

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