High School

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

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 :

We start with the equation

[tex]$$
4x^4 - 21x^2 + 20 = 0.
$$[/tex]

Notice that the highest power is [tex]$x^4$[/tex], but it can be expressed as [tex]$(x^2)^2$[/tex]. This observation suggests making the substitution

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

Then, we have

[tex]$$
x^4 = (x^2)^2 = u^2.
$$[/tex]

Substituting into the original equation yields

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

This is a quadratic equation in [tex]$u$[/tex]. Thus, the correct substitution is

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

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