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, we choose a suitable substitution.

Let's consider the substitution [tex]\(u = x^2\)[/tex].

Here's why this works:

1. Identify expressions with [tex]\(x\)[/tex]: In the original equation, we see terms [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].

2. Relate [tex]\(u\)[/tex] to these terms:
- Since [tex]\(u = x^2\)[/tex], it follows that [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].

3. Rewrite the equation: Substituting [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex], we get:
[tex]\[
4x^4 - 21x^2 + 20 = 0
\][/tex]
becomes
[tex]\[
4(u^2) - 21u + 20 = 0
\][/tex]

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

Therefore, the correct substitution to rewrite the original equation as a quadratic is [tex]\(u = x^2\)[/tex].