To solve the quadratic equation $x^2 + 4x - 12 = 0$ using the quadratic formula, we can follow these steps:

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a$, $b$, and $c$ are the coefficients from the general quadratic equation $ax^2 + bx + c = 0$.

For the equation $x^2 + 4x - 12 = 0$:

• $a = 1$
• $b = 4$
• $c = -12$

Now, plug these values into the quadratic formula:

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)}$ Simplify inside the square root: $x = \frac{-4 \pm \sqrt{16 + 48}}{2}$ $x = \frac{-4 \pm \sqrt{64}}{2}$ $x = \frac{-4 \pm 8}{2}$

Now, solve for the two possible values of $x$:

1. $x_1 = \frac{-4 + 8}{2} = \frac{4}{2} = 2$
2. $x_2 = \frac{-4 - 8}{2} = \frac{-12}{2} = -6$

Thus, the solutions are: $x = 2 \quad \text{or} \quad x = -6$

Would you like a detailed explanation of any step or have any questions?

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Here are 5 related questions:

1. How can you factor a quadratic equation instead of using the quadratic formula?
2. What are the key differences between real and complex solutions in quadratics?
3. How does the discriminant $b^2 - 4ac$ affect the nature of the solutions?
4. Can every quadratic equation be solved using the quadratic formula?
5. What are the geometric interpretations of quadratic solutions on a graph?

Tip: Always check the discriminant first to determine if the solutions will be real or complex!