Solve for all values of $$ x $$ by factoring $$ x^{2}+x-12=x+4 $$

Question

UpStudy AI · Accepted Answer

To solve the equation

$$
x^{2} + x - 12 = x + 4
$$

by factoring, follow these steps:

1. **Rearrange the Equation:**

First, bring all terms to one side to set the equation to zero:

$$
   x^{2} + x - 12 - x - 4 = 0
   $$

Simplifying this, we get:

$$
   x^{2} - 16 = 0
   $$

2. **Factor the Quadratic:**

The equation $$ x^{2} - 16 = 0 $$ is a difference of squares, which can be factored as:

$$
   (x - 4)(x + 4) = 0
   $$

3. **Find the Solutions:**

Set each factor equal to zero and solve for $$ x $$:

$$
   x - 4 = 0 \quad \Rightarrow \quad x = 4
   $$

$$
   x + 4 = 0 \quad \Rightarrow \quad x = -4
   $$

4. **Verify the Solutions:**

- **For $$ x = 4 $$:**

$$
     \text{LHS} = 4^{2} + 4 - 12 = 16 + 4 - 12 = 8
     $$
     $$
     \text{RHS} = 4 + 4 = 8
     $$
     Both sides are equal.

- **For $$ x = -4 $$:**

$$
     \text{LHS} = (-4)^{2} + (-4) - 12 = 16 - 4 - 12 = 0
     $$
     $$
     \text{RHS} = -4 + 4 = 0
     $$
     Both sides are equal.

Both solutions are valid.

**Final Answer:**
$$
x =\,4 \quad \text{and} \quad x =\, -4
$$
$$ x = 4 $$ and $$ x = -4 $$

Solve for all values of \( x \) by factoring \[ x^{2}+x-12=x+4 \]