The problem asks us to solve the equation $x^2 = \frac{6}{7}x$ using the quadratic formula and simplify the solution.

1. Problem Description

The problem asks us to solve the equation $x^2 = \frac{6}{7}x$ using the quadratic formula and simplify the solution.

2. Solution Steps

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

$x^2 = \frac{6}{7}x$

$x^2 - \frac{6}{7}x = 0$

Now, we have $a = 1$, $b = -\frac{6}{7}$, and $c = 0$. The quadratic formula is given by:

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

Plugging in the values for $a$, $b$, and $c$:

$x = \frac{-(-\frac{6}{7}) \pm \sqrt{(-\frac{6}{7})^2 - 4(1)(0)}}{2(1)}$

$x = \frac{\frac{6}{7} \pm \sqrt{\frac{36}{49} - 0}}{2}$

$x = \frac{\frac{6}{7} \pm \sqrt{\frac{36}{49}}}{2}$

$x = \frac{\frac{6}{7} \pm \frac{6}{7}}{2}$

Now we have two possible solutions:

$x_1 = \frac{\frac{6}{7} + \frac{6}{7}}{2} = \frac{\frac{12}{7}}{2} = \frac{12}{7} \cdot \frac{1}{2} = \frac{6}{7}$

$x_2 = \frac{\frac{6}{7} - \frac{6}{7}}{2} = \frac{0}{2} = 0$

So the solutions are $x = 0$ and $x = \frac{6}{7}$.

3. Final Answer

0, 6/7