The problem asks to solve the quadratic equation $4x^2 - x - 3 = 0$ using the quadratic formula.

1. Problem Description

The problem asks to solve the quadratic equation $4x^2 - x - 3 = 0$ using the quadratic formula.

2. Solution Steps

The quadratic formula is given by:

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

For the equation $4x^2 - x - 3 = 0$, we have $a = 4$, $b = -1$, and $c = -3$.

Substitute these values into the quadratic formula:

$$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-3)}}{2(4)}$$

$$x = \frac{1 \pm \sqrt{1 + 48}}{8}$$

$$x = \frac{1 \pm \sqrt{49}}{8}$$

$$x = \frac{1 \pm 7}{8}$$

So, the two possible values for $x$ are:

$$x_1 = \frac{1 + 7}{8} = \frac{8}{8} = 1$$

$$x_2 = \frac{1 - 7}{8} = \frac{-6}{8} = -\frac{3}{4}$$

3. Final Answer

$x = 1, -\frac{3}{4}$