Solve the quadratic by factoring.

x, squared, minus, 22, equals, 5, x, plus, 2

*x*2−22=5*x*+2

Let's solve the quadratic equation \( x^2 - 22 = 5x + 2 \) by factoring.

### Step 1: Move all terms to one side of the equation.
To do this, subtract \( 5x + 2 \) from both sides:

\[
x^2 - 22 - 5x - 2 = 0
\]

Simplify the equation:

\[
x^2 - 5x - 24 = 0
\]

### Step 2: Factor the quadratic expression.
We need to find two numbers that multiply to \(-24\) (the constant term) and add to \(-5\) (the coefficient of \(x\)).

The two numbers are \(-8\) and \(3\), because:

\[
-8 \times 3 = -24 \quad \text{and} \quad -8 + 3 = -5
\]

So, we can factor the quadratic as:

\[
(x - 8)(x + 3) = 0
\]

### Step 3: Solve for \(x\).
Set each factor equal to zero:

\[
x - 8 = 0 \quad \text{or} \quad x + 3 = 0
\]

Solve each equation:

\[
x = 8 \quad \text{or} \quad x = -3
\]

### Final Answer:
The solutions to the equation \( x^2 - 22 = 5x + 2 \) are \( x = 8 \) and \( x = -3 \).

Would you like any details or have further questions?

Here are some related questions:
1. How do you factor quadratics when the leading coefficient isn't 1?
2. What is the quadratic formula and when should it be used?
3. How can you complete the square to solve a quadratic equation?
4. What is the discriminant, and how does it help determine the nature of the roots?
5. Can all quadratic equations be factored?

**Tip:** Always check your solutions by substituting them back into the original equation.

Solve the quadratic by factoring.

x squared minus 22 equals 5x plus 2

x^2−22=5x+2

Learn how to solve the quadratic equation x^2 - 22 = 5x + 2 by factoring. This step-by-step guide covers essential algebraic concepts, including factoring and the Zero Product Property. Ideal for students in Grades 8-10.

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