The problem is to solve the quadratic equation $2x^2 + 4x - 6 = 0$.

1. Problem Description

The problem is to solve the quadratic equation $2x^2 + 4x - 6 = 0$.

2. Solution Steps

First, we can simplify the equation by dividing all terms by 2:

$x^2 + 2x - 3 = 0$

Next, we can factor the quadratic expression:

We are looking for two numbers that multiply to -3 and add up to

2. Those numbers are 3 and -

1.

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

Now, we set each factor to zero and solve for $x$:

$x + 3 = 0$ or $x - 1 = 0$

$x = -3$ or $x = 1$

Alternatively, we can use the quadratic formula:

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

For the simplified equation $x^2 + 2x - 3 = 0$, $a = 1$, $b = 2$, and $c = -3$.

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

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

$x = \frac{-2 \pm \sqrt{16}}{2}$

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

$x = \frac{-2 + 4}{2}$ or $x = \frac{-2 - 4}{2}$

$x = \frac{2}{2}$ or $x = \frac{-6}{2}$

$x = 1$ or $x = -3$

3. Final Answer

The solutions are $x = 1$ and $x = -3$.