Question

what substitution should be used to rewrite $4x^4 - 21x^2 + 20 = 0$ as a quadratic equation?
$\bigcirc$ $u = x^2$
$\bigcirc$ $u = 2x^2$
$\bigcirc$ $u = x^4$
$\bigcirc$ $u = 4x^4$

Explanation:

Step1: Recall Quadratic Form

A quadratic equation is of the form \(au^2 + bu + c = 0\), where \(u\) is a variable. The given equation is \(4x^4 - 21x^2 + 20 = 0\). Notice that \(x^4=(x^2)^2\).

Step2: Test Substitutions

• For \(u = x^2\), substitute into the equation: \(4u^2 - 21u + 20 = 0\), which is a quadratic equation.
• For \(u = 2x^2\), substitution gives \(4(\frac{u}{2})^2 - 21(\frac{u}{2}) + 20 = u^2 - \frac{21}{2}u + 20\), not a simple quadratic.
• For \(u = x^4\), substitution gives \(4u - 21\sqrt{u} + 20 = 0\) (since \(x^2=\sqrt{u}\)), not quadratic.
• For \(u = 4x^4\), substitution gives \(u - 21x^2 + 20 = 0\), not quadratic.

Answer:

A. \(u = x^2\)