Question
Solve for all values of xx by factoring.
x, squared, minus, 10, x, plus, 21, equals, x, minus, 3
x
2
−10x+21=
x−3
x, squared, minus, 10, x, plus, 21, equals, x, minus, 3
x
2
−10x+21=
x−3
Answers
GPT-4o mini
To solve for \( x \) in the equation:
\[
x^2 - 10x + 21 = x - 3
\]
First, we will rearrange the equation to set it to zero:
\[
x^2 - 10x + 21 - x + 3 = 0
\]
Simplifying the left-hand side, we have:
\[
x^2 - 11x + 24 = 0
\]
Next, we need to factor the quadratic expression \( x^2 - 11x + 24 \). We are looking for two numbers that multiply to \( 24 \) (the constant term) and add up to \( -11 \) (the coefficient of \( x \)). The numbers that meet these criteria are \( -3 \) and \( -8 \).
Thus, we can factor the quadratic as:
\[
(x - 3)(x - 8) = 0
\]
Now, we set each factor to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
Therefore, the solutions to the equation are:
\[
\boxed{3 \text{ and } 8}
\]
\[
x^2 - 10x + 21 = x - 3
\]
First, we will rearrange the equation to set it to zero:
\[
x^2 - 10x + 21 - x + 3 = 0
\]
Simplifying the left-hand side, we have:
\[
x^2 - 11x + 24 = 0
\]
Next, we need to factor the quadratic expression \( x^2 - 11x + 24 \). We are looking for two numbers that multiply to \( 24 \) (the constant term) and add up to \( -11 \) (the coefficient of \( x \)). The numbers that meet these criteria are \( -3 \) and \( -8 \).
Thus, we can factor the quadratic as:
\[
(x - 3)(x - 8) = 0
\]
Now, we set each factor to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
Therefore, the solutions to the equation are:
\[
\boxed{3 \text{ and } 8}
\]