The problem is to find all the roots of the quadratic equation $5x^2 = 6x - 1$.

1. Problem Description

The problem is to find all the roots of the quadratic equation $5x^2 = 6x - 1$.

2. Solution Steps

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

$5x^2 = 6x - 1$

$5x^2 - 6x + 1 = 0$

Now, we can use the quadratic formula to find the roots of the equation. The quadratic formula is:

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

In our case, $a = 5$, $b = -6$, and $c = 1$. Plugging these values into the quadratic formula:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(5)(1)}}{2(5)}$

$x = \frac{6 \pm \sqrt{36 - 20}}{10}$

$x = \frac{6 \pm \sqrt{16}}{10}$

$x = \frac{6 \pm 4}{10}$

Now we have two possible solutions:

$x_1 = \frac{6 + 4}{10} = \frac{10}{10} = 1$

$x_2 = \frac{6 - 4}{10} = \frac{2}{10} = \frac{1}{5}$

3. Final Answer

The roots are $1, \frac{1}{5}$.