To solve the quadratic equation \( x^2 + 2x - 12 = 0 \), we can either factor the equation or use the quadratic formula. Let's go step by step with both methods:

### Method 1: Factoring

1. We want to express \( x^2 + 2x - 12 \) as a product of two binomials. The general form will be:
   \[
   (x + p)(x + q) = 0
   \]
   We need to find \( p \) and \( q \) such that:
   - \( p + q = 2 \) (the coefficient of \( x \))
   - \( p \cdot q = -12 \) (the constant term)

2. After trying different factors of \(-12\), we find:
   \[
   p = 4 \quad \text{and} \quad q = -3
   \]
   Since \( 4 + (-3) = 1 \) and \( 4 \cdot (-3) = -12 \), the factored form is:
   \[
   (x + 4)(x - 3) = 0
   \]

3. Now, solve for \( x \):
   - \( x + 4 = 0 \Rightarrow x = -4 \)
   - \( x - 3 = 0 \Rightarrow x = 3 \)

So, the solutions are:
\[
x = -4 \quad \text{or} \quad x = 3
\]

### Method 2: Quadratic Formula

The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( x^2 + 2x - 12 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 2 \)
- \( c = -12 \)

Substitute these values into the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-12)}}{2(1)}
\]
\[
x = \frac{-2 \pm \sqrt{4 + 48}}{2}
\]
\[
x = \frac{-2 \pm \sqrt{52}}{2}
\]
\[
x = \frac{-2 \pm 2\sqrt{13}}{2}
\]
\[
x = -1 \pm \sqrt{13}
\]
So the solutions are:
\[
x = -1 + \sqrt{13} \quad \text{or} \quad x = -1 - \sqrt{13}
\]

Both methods give equivalent solutions (factoring gives exact integers while the quadratic formula provides exact or approximate answers involving radicals).

Would you like more details on either method?

### 5 Related Questions:
1. What is the discriminant, and how does it help in solving quadratic equations?
2. How can we graph the equation \( x^2 + 2x - 12 = 0 \)?
3. Can all quadratic equations be factored easily?
4. What is the difference between exact and approximate solutions in quadratic equations?
5. How does completing the square work for solving quadratic equations?

**Tip**: Always check if a quadratic equation can be factored first, as it may provide simpler solutions than using the quadratic formula.

Explore two methods for solving the quadratic equation x^2 + 2x - 12: factoring and using the quadratic formula. This solution breaks down the steps and provides clear explanations, ideal for students in Grades 9-10.

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