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

1. Problem Description

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

2. Solution Steps

The quadratic formula is given by:

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

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

In our case, $a = 2$, $b = -1$, and $c = -3$.

Substituting these values into the quadratic formula, we get:

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

$x = \frac{1 \pm \sqrt{1 + 24}}{4}$

$x = \frac{1 \pm \sqrt{25}}{4}$

$x = \frac{1 \pm 5}{4}$

So we have two possible solutions:

$x_1 = \frac{1 + 5}{4} = \frac{6}{4} = \frac{3}{2}$

$x_2 = \frac{1 - 5}{4} = \frac{-4}{4} = -1$

Therefore, the solutions are $x = \frac{3}{2}$ and $x = -1$.

3. Final Answer

$\frac{3}{2}, -1$