what are the solutions of $x^{2}=8 - 5x$?\n$\frac{5-sqrt{57}}{2},\frac{5 + sqrt{57}}{2}$\n$\frac{-5-sqrt{57}}…

Explanation:

Step1: Rearrange to standard quadratic form

First, rewrite the equation $x^{2}=8 - 5x$ as $x^{2}+5x - 8=0$. The general form of a quadratic equation is $ax^{2}+bx + c = 0$, where here $a = 1$, $b = 5$, and $c=-8$.

Step2: Apply quadratic formula

The quadratic formula for the solutions of $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute $a = 1$, $b = 5$, and $c=-8$ into the formula: [ \begin{align*} x&=\frac{-5\pm\sqrt{5^{2}-4\times1\times(-8)}}{2\times1}\ &=\frac{-5\pm\sqrt{25 + 32}}{2}\ &=\frac{-5\pm\sqrt{57}}{2} \end{align*} ] The two solutions are $x=\frac{-5-\sqrt{57}}{2}$ and $x=\frac{-5+\sqrt{57}}{2}$.

Answer:

$\frac{-5 - \sqrt{57}}{2},\frac{-5+\sqrt{57}}{2}$