College

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 can use a substitution method. Here’s a step-by-step explanation:

1. Identify the structure of the equation: The equation is [tex]\(4x^4 - 21x^2 + 20 = 0\)[/tex]. Notice that the terms involve powers of [tex]\(x\)[/tex] that are multiples of 2, specifically [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex].

2. Choose a substitution: To transform this into a standard quadratic form, we can use the substitution [tex]\( u = x^2 \)[/tex]. This is because [tex]\(x^4\)[/tex] can be expressed as [tex]\((x^2)^2\)[/tex], which is [tex]\(u^2\)[/tex].

3. Rewrite the equation: Replace [tex]\(x^4\)[/tex] with [tex]\(u^2\)[/tex] and [tex]\(x^2\)[/tex] with [tex]\(u\)[/tex] in the original equation:

[tex]\[
4(x^2)^2 - 21(x^2) + 20 = 0
\][/tex]

Using the substitution [tex]\(u = x^2\)[/tex], this becomes:

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

4. Resulting quadratic equation: After substitution, the equation is [tex]\(4u^2 - 21u + 20 = 0\)[/tex], which is a standard quadratic equation in terms of [tex]\(u\)[/tex].

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