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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]

To rewrite the equation [tex]$4x^4 - 21x^2 + 20 = 0$[/tex] as a quadratic equation, follow these steps:

1. Identify the Target Form: We want to transform the given equation into a standard quadratic form, which is [tex]$au^2 + bu + c = 0$[/tex], where [tex]$u$[/tex] is some expression involving [tex]$x$[/tex].

2. Choose the Substitution: Notice that the given equation has terms [tex]$x^4$[/tex] and [tex]$x^2$[/tex]. If we let [tex]$u = x^2$[/tex], then:

- [tex]$x^4$[/tex] can be rewritten as [tex]$(x^2)^2$[/tex], which is [tex]$u^2$[/tex].

3. Rewrite the Equation: Substitute [tex]$u = x^2$[/tex] into the given equation.

- [tex]$4x^4$[/tex] becomes [tex]$4(u^2)$[/tex].
- [tex]$-21x^2$[/tex] becomes [tex]$-21u$[/tex].

Now, the equation [tex]$4x^4 - 21x^2 + 20 = 0$[/tex] transforms into:

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

4. Result: The equation is now a quadratic in [tex]$u$[/tex]: [tex]$4u^2 - 21u + 20 = 0$[/tex].

The appropriate substitution to rewrite the original equation as a quadratic equation is [tex]$u = x^2$[/tex].