Certainly! Let’s go through the steps needed to solve the quadratic equation [tex]\( x^2 + 4x - 12 = 0 \)[/tex] by factoring.
### Step 1: Write down the given quadratic equation
We start with:
[tex]\[ x^2 + 4x - 12 = 0 \][/tex]
### Step 2: Find two numbers that multiply to the constant term (-12) and add up to the coefficient of the linear term (4)
We need to identify two numbers such that:
[tex]\[ \text{Product} = -12 \][/tex]
[tex]\[ \text{Sum} = 4 \][/tex]
The two numbers that meet these criteria are 6 and -2 because:
[tex]\[ 6 \cdot (-2) = -12 \][/tex]
[tex]\[ 6 + (-2) = 4 \][/tex]
### Step 3: Rewrite the quadratic equation using these two numbers
Using the numbers 6 and -2, we can factor the quadratic equation as follows:
[tex]\[
x^2 + 4x - 12 = (x + 6)(x - 2)
\][/tex]
### Step 4: Set each factor equal to zero and solve for [tex]\( x \)[/tex]
Now, we solve for [tex]\( x \)[/tex] by setting each factor equal to zero:
1. For the factor [tex]\( x + 6 \)[/tex]:
[tex]\[
x + 6 = 0
\][/tex]
[tex]\[
x = -6
\][/tex]
2. For the factor [tex]\( x - 2 \)[/tex]:
[tex]\[
x - 2 = 0
\][/tex]
[tex]\[
x = 2
\][/tex]
### Conclusion
The solutions to the quadratic equation [tex]\( x^2 + 4x - 12 = 0 \)[/tex] are:
[tex]\[ x = -6 \quad \text{and} \quad x = 2 \][/tex]
In summary:
- The factored form of the quadratic is [tex]\( (x + 6)(x - 2) \)[/tex].
- The solutions are [tex]\( x = -6 \)[/tex] and [tex]\( x = 2 \)[/tex].
So, the complete factored form and the solutions are:
[tex]\[
(x + 6)(x - 2) = 0
\][/tex]
[tex]\[
x = -6, \quad x = 2
\][/tex]
