Question

what substitution should be used to rewrite $x^8 - 3x^4 + 2 = 0$ as a quadratic equation?
$\circ$ $u = x^2$
$\circ$ $u = x^4$
$\circ$ $u = x^8$
$\circ$ $u = x^{16}$

Explanation:

Step1: Recall Quadratic Form

A quadratic equation is of the form \(au^2 + bu + c = 0\). We need to express \(x^8 - 3x^4 + 2 = 0\) in such a form.

Step2: Analyze Exponents

Notice that \(x^8=(x^4)^2\). Let \(u = x^4\). Then \(x^8 = u^2\) and the equation becomes \(u^2 - 3u + 2 = 0\), which is quadratic.

Step3: Check Other Options

• If \(u = x^2\), \(x^8=(x^2)^4 = u^4\), so equation is \(u^4 - 3u^2 + 2 = 0\) (not quadratic).
• If \(u = x^8\), equation is \(u - 3x^4 + 2 = 0\) (not quadratic in \(u\)).
• If \(u = x^{16}\), equation is \(x^8 - 3x^4 + 2 = 0\) (no relation to \(u\), not quadratic).

Answer:

B. \(u = x^4\)