Question

solve the equation. if necessary, round to the nearest hundredth.\\(x^4 - 15x^2 = -24\\)\\(\bigcirc\\) a) \\(-3.63, -1.35, 3.63\\)\\(\bigcirc\\) b) \\(-3.63, 1.35, 3.63\\)\\(\bigcirc\\) c) \\(-3.63, -1.35, 1.35, 3.63\\)\\(\bigcirc\\) d) no solution

Explanation:

Step1: Rewrite to standard quartic form

$x^4 - 15x^2 + 24 = 0$

Step2: Substitute $y=x^2$

$y^2 - 15y + 24 = 0$

Step3: Solve quadratic for $y$

Use quadratic formula $y=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=-15, c=24$

$$\begin{align*} y&=\frac{15\pm\sqrt{(-15)^2-4(1)(24)}}{2(1)}\\ &=\frac{15\pm\sqrt{225-96}}{2}\\ &=\frac{15\pm\sqrt{129}}{2} \end{align*}$$

Step4: Calculate $y$ values

$y_1=\frac{15+\sqrt{129}}{2}\approx\frac{15+11.36}{2}\approx13.18$, $y_2=\frac{15-\sqrt{129}}{2}\approx\frac{15-11.36}{2}\approx1.82$

Step5: Solve for $x$ from $y=x^2$

For $y_1\approx13.18$: $x=\pm\sqrt{13.18}\approx\pm3.63$
For $y_2\approx1.82$: $x=\pm\sqrt{1.82}\approx\pm1.35$

Answer:

C) -3.63, -1.35, 1.35, 3.63