High School

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 make the substitution

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

Notice that with this substitution, we have

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

Substituting into the equation gives

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

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

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