Solve for all values of xx by factoring.

Question

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:

$$-2\cdot -9=18\text{and}-2+-9=-11$$

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:

$$\boxed{{\displaystyle 2\text{ and }9}}$$