The problem is to solve the quadratic equation $2x(x+2) - 8 = 2x - 7$ using the quadratic formula.

1. Problem Description

The problem is to solve the quadratic equation $2x(x+2) - 8 = 2x - 7$ using the quadratic formula.

2. Solution Steps

First, expand and simplify the equation:

$2x(x+2) - 8 = 2x - 7$

$2x^2 + 4x - 8 = 2x - 7$

$2x^2 + 4x - 2x - 8 + 7 = 0$

$2x^2 + 2x - 1 = 0$

Now, we have a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 2$, $b = 2$, and $c = -1$.

We can use the quadratic formula to solve for $x$:

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

Plugging in the values of $a$, $b$, and $c$:

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

$x = \frac{-2 \pm \sqrt{4 + 8}}{4}$

$x = \frac{-2 \pm \sqrt{12}}{4}$

$x = \frac{-2 \pm 2\sqrt{3}}{4}$

$x = \frac{-1 \pm \sqrt{3}}{2}$

Thus, the two solutions are:

$x = \frac{-1 + \sqrt{3}}{2}$ and $x = \frac{-1 - \sqrt{3}}{2}$

3. Final Answer

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