The problem asks us to find all the roots of the equation $15x^2 - 3 = 4x$.

1. Problem Description

The problem asks us to find all the roots of the equation $15x^2 - 3 = 4x$.

2. Solution Steps

First, rearrange the equation to the standard quadratic form $ax^2 + bx + c = 0$:

$15x^2 - 4x - 3 = 0$

Now, we can use the quadratic formula to find the roots:

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

In our case, $a = 15$, $b = -4$, and $c = -3$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(15)(-3)}}{2(15)}$

$x = \frac{4 \pm \sqrt{16 + 180}}{30}$

$x = \frac{4 \pm \sqrt{196}}{30}$

$x = \frac{4 \pm 14}{30}$

So, the two roots are:

$x_1 = \frac{4 + 14}{30} = \frac{18}{30} = \frac{3}{5}$

$x_2 = \frac{4 - 14}{30} = \frac{-10}{30} = -\frac{1}{3}$

3. Final Answer

The roots are $\frac{3}{5}, -\frac{1}{3}$.