The problem asks us to solve the quadratic equation $x^2 = -\frac{5}{6}x$ using the quadratic formula and simplify the answer.

1. Problem Description

The problem asks us to solve the quadratic equation $x^2 = -\frac{5}{6}x$ using the quadratic formula and simplify the answer.

2. Solution Steps

First, we rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$.

$x^2 = -\frac{5}{6}x$

$x^2 + \frac{5}{6}x = 0$

Now we can multiply the entire equation by 6 to get rid of the fraction:

$6(x^2 + \frac{5}{6}x) = 6(0)$

$6x^2 + 5x = 0$

Thus, we have $a = 6$, $b = 5$, and $c = 0$.

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting the values of $a$, $b$, and $c$, we have:

$x = \frac{-5 \pm \sqrt{5^2 - 4(6)(0)}}{2(6)}$

$x = \frac{-5 \pm \sqrt{25 - 0}}{12}$

$x = \frac{-5 \pm \sqrt{25}}{12}$

$x = \frac{-5 \pm 5}{12}$

We have two possible solutions:

$x_1 = \frac{-5 + 5}{12} = \frac{0}{12} = 0$

$x_2 = \frac{-5 - 5}{12} = \frac{-10}{12} = -\frac{5}{6}$

3. Final Answer

0, -5/6