Question

Solve for all values of xx by factoring.
x, squared, minus, 20, x, plus, 90, equals, minus, x
x
2
−20x+90=
−x

Answer 1

To solve the equation \( x^2 - 20x + 90 = -x \), we first bring all terms to one side of the equation to set it to zero.

Starting with the original equation:

\[
x^2 - 20x + 90 + x = 0
\]

This simplifies to:

\[
x^2 - 19x + 90 = 0
\]

Next, we need to factor the quadratic equation \( x^2 - 19x + 90 \). We are looking for two numbers that multiply to \( 90 \) (the constant term) and add to \( -19 \) (the coefficient of \( x \)). The numbers \( -9 \) and \( -10 \) meet these criteria:

\[
-9 \cdot -10 = 90 \quad \text{and} \quad -9 + -10 = -19
\]

So, we can factor the quadratic as follows:

\[
(x - 9)(x - 10) = 0
\]

Setting each factor equal to zero gives us the possible solutions for \( x \):

1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x - 10 = 0 \) → \( x = 10 \)

Therefore, the solutions are:

\[
\boxed{9 \text{ and } 10}
\]