To solve the quadratic equation $x^2 + 2x - 12 = 0$, we can use the quadratic formula:

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

In this case, the coefficients are:

• $a = 1$ (coefficient of $x^2$),
• $b = 2$ (coefficient of $x$),
• $c = -12$ (constant term).

Step 1: Calculate the discriminant

$\Delta = b^2 - 4ac = 2^2 - 4(1)(-12) = 4 + 48 = 52$

Step 2: Apply the quadratic formula

$x = \frac{-2 \pm \sqrt{52}}{2(1)} = \frac{-2 \pm \sqrt{52}}{2}$ Simplifying $\sqrt{52} = 2\sqrt{13}$: $x = \frac{-2 \pm 2\sqrt{13}}{2}$

Step 3: Simplify the expression

$x = -1 \pm \sqrt{13}$

Thus, the solutions are: $x_1 = -1 + \sqrt{13}, \quad x_2 = -1 - \sqrt{13}$

Would you like further details or clarification on any step?

Here are some related questions to practice:

1. Solve $x^2 - 6x + 8 = 0$.
2. Solve $2x^2 + 3x - 5 = 0$ using the quadratic formula.
3. Factor the quadratic equation $x^2 - 9x + 20 = 0$.
4. Find the roots of $x^2 - 4 = 0$.
5. Determine the discriminant for the equation $3x^2 + 5x + 2 = 0$.

Tip: Always check the discriminant ($b^2 - 4ac$) to determine the nature of the roots (real or complex).