Question

Solve for all values of xx by factoring.
x 2 −10x+21=x−3

Answer 1

To solve the equation \( x^2 - 10x + 21 = x - 3 \) by factoring, we first rearrange the equation to set it to zero:

\[
x^2 - 10x + 21 - x + 3 = 0
\]

Combine like terms:

\[
x^2 - 11x + 24 = 0
\]

Next, we need to factor the quadratic expression \( x^2 - 11x + 24 \). We look for two numbers that multiply to \( 24 \) (the constant term) and add to \( -11 \) (the coefficient of \( x \)). These numbers are \( -3 \) and \( -8 \). Therefore, we can factor the expression as follows:

\[
(x - 3)(x - 8) = 0
\]

Now, we can set each factor equal to zero:

1. \( x - 3 = 0 \)
- This gives \( x = 3 \).

2. \( x - 8 = 0 \)
- This gives \( x = 8 \).

Thus, the solutions to the equation \( x^2 - 10x + 21 = x - 3 \) are:

\[
\boxed{3} \quad \text{and} \quad \boxed{8}
\]