The problem asks us to find the roots of the quadratic equation $12x^2 - 5 = 17x$. We need to express the answer as an integer or a fraction, separating multiple roots by commas.

1. Problem Description

The problem asks us to find the roots of the quadratic equation $12x^2 - 5 = 17x$. We need to express the answer as an integer or a fraction, separating multiple roots by commas.

2. Solution Steps

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

$12x^2 - 17x - 5 = 0$

We can solve this quadratic equation by factoring. We look for two numbers that multiply to $12 \times -5 = -60$ and add up to $-17$. These numbers are $-20$ and $3$. We rewrite the middle term using these two numbers:

$12x^2 - 20x + 3x - 5 = 0$

Now we factor by grouping:

$4x(3x - 5) + 1(3x - 5) = 0$

$(4x + 1)(3x - 5) = 0$

Setting each factor to zero gives us the two possible values for $x$:

$4x + 1 = 0 \Rightarrow 4x = -1 \Rightarrow x = -\frac{1}{4}$

$3x - 5 = 0 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3}$

Therefore, the roots are $-\frac{1}{4}$ and $\frac{5}{3}$.

3. Final Answer

$-\frac{1}{4}, \frac{5}{3}$