Question
Solve for all values of xx by factoring.
x, squared, minus, 10, x, plus, 18, equals, x
x
2
−10x+18=
x
x, squared, minus, 10, x, plus, 18, equals, x
x
2
−10x+18=
x
Answers
GPT-4o mini
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:
\[
\boxed{2 \text{ and } 9}
\]
\[
x^2 - 10x + 18 - x = 0
\]
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:
\[
\boxed{2 \text{ and } 9}
\]