Solve for all values of $$ x $$ by factoring. $$ x^{2}-13 x+24=-x+4 $$

Question

UpStudy AI · Accepted Answer

To solve the equation by factoring:

$$
x^{2} - 13x + 24 = -x + 4
$$

**Step 1: Move all terms to one side of the equation**

Add $$ x $$ and subtract $$ 4 $$ from both sides:

$$
x^{2} - 13x + 24 + x - 4 = 0
$$

Simplify:

$$
x^{2} - 12x + 20 = 0
$$

**Step 2: Factor the quadratic equation**

We need to find two numbers that multiply to $$ 20 $$ and add up to $$ -12 $$.

The numbers $$ -10 $$ and $$ -2 $$ satisfy these conditions:

$$
-10 \times -2 = 20 \quad \text{and} \quad -10 + (-2) = -12
$$

So, the factored form is:

$$
(x - 10)(x - 2) = 0
$$

**Step 3: Solve for $$ x $$**

Set each factor equal to zero:

$$
x - 10 = 0 \quad \Rightarrow \quad x = 10
$$
$$
x - 2 = 0 \quad \Rightarrow \quad x = 2
$$

**Solution:**

The values of $$ x $$ that satisfy the equation are $$ x = 10 $$ and $$ x = 2 $$.

**Final Answer:** 
$$\boxed{\left\{2,\;10\,\right\}}$$
The solutions are $$ x = 2 $$ and $$ x = 10 $$.

Solve for all values of \( x \) by factoring. \[ x^{2}-13 x+24=-x+4 \]