Question

Solve for all values of xx by factoring.
x, squared, minus, 10, x, plus, 18, equals, x
x
2
−10x+18=
x

Answer 1

To solve the equation \( x^2 - 10x + 18 = x \) by factoring, we first move all terms to one side of the equation:

\[
x^2 - 10x + 18 - x = 0
\]

This simplifies to:

\[
x^2 - 11x + 18 = 0
\]

Next, we will factor the quadratic expression \( x^2 - 11x + 18 \).

We are looking for two numbers that multiply to \( 18 \) (the constant term) and add to \( -11 \) (the coefficient of \( x \)). The numbers \( -2 \) and \( -9 \) satisfy both conditions:

\[
-2 \cdot -9 = 18 \quad \text{and} \quad -2 + -9 = -11
\]

Thus, we can factor the quadratic as follows:

\[
(x - 2)(x - 9) = 0
\]

Setting each factor equal to zero gives us the possible solutions:

1. \( x - 2 = 0 \) which leads to \( x = 2 \)
2. \( x - 9 = 0 \) which leads to \( x = 9 \)

Therefore, the solutions to the equation \( x^2 - 10x + 18 = x \) are:

\[
\boxed{2 \text{ and } 9}
\]