We need to solve the quadratic equation $7x^2 - 6x - 3 = 0$ for $x$.

1. Problem Description

We need to solve the quadratic equation $7x^2 - 6x - 3 = 0$ for $x$.

2. Solution Steps

We can solve this quadratic equation using the quadratic formula.

The quadratic formula is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

In our case, $a = 7$, $b = -6$, and $c = -3$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(7)(-3)}}{2(7)}$

$x = \frac{6 \pm \sqrt{36 + 84}}{14}$

$x = \frac{6 \pm \sqrt{120}}{14}$

$x = \frac{6 \pm \sqrt{4 \cdot 30}}{14}$

$x = \frac{6 \pm 2\sqrt{30}}{14}$

$x = \frac{3 \pm \sqrt{30}}{7}$

3. Final Answer

$x = \frac{3 + \sqrt{30}}{7}$ and $x = \frac{3 - \sqrt{30}}{7}$