Question

Solve for all values of xx by factoring.
x, squared, minus, 2, x, minus, 45, equals, 2, x
x
2
−2x−45=
2x

Answer 1

To solve the equation \( x^2 - 2x - 45 = 2x \) by factoring, we first need to rearrange the equation so that one side is equal to zero.

### Step 1: Move all terms to one side
Subtract \( 2x \) from both sides of the equation:

\[
x^2 - 2x - 45 - 2x = 0
\]

This simplifies to:

\[
x^2 - 4x - 45 = 0
\]

### Step 2: Factor the quadratic
Next, we need to factor the quadratic expression \( x^2 - 4x - 45 \).

To factor, we need two numbers that multiply to \(-45\) (the constant term) and add to \(-4\) (the coefficient of \(x\)). The numbers that fit this are \(5\) and \(-9\) because:

\[
5 \times (-9) = -45
\]
\[
5 + (-9) = -4
\]

So we can write the factored form as:

\[
(x - 9)(x + 5) = 0
\]

### Step 3: Solve for \(x\)
Now we set each factor equal to zero:

1. \( x - 9 = 0 \)
\[
x = 9
\]

2. \( x + 5 = 0 \)
\[
x = -5
\]

### Conclusion
The solutions to the equation \( x^2 - 2x - 45 = 2x \) are:

\[
\boxed{9} \quad \text{and} \quad \boxed{-5}
\]