Question
Solve for all values of xx by factoring.
x, squared, plus, 21, x, plus, 50, equals, 6, x
x
2
+21x+50=
6x
x, squared, plus, 21, x, plus, 50, equals, 6, x
x
2
+21x+50=
6x
Answers
GPT-4o mini
To solve the equation \( x^2 + 21x + 50 = 6x \) by factoring, we start by rearranging the equation so that all terms are on one side:
\[
x^2 + 21x + 50 - 6x = 0
\]
This simplifies to:
\[
x^2 + 15x + 50 = 0
\]
Next, we will factor the quadratic expression \( x^2 + 15x + 50 \). We need to find two numbers that multiply to 50 (the constant term) and add up to 15 (the coefficient of the linear term).
The pairs of factors of 50 are:
- \( 1 \times 50 \)
- \( 2 \times 25 \)
- \( 5 \times 10 \)
Among these, the pair \( 5 \) and \( 10 \) adds up to \( 15 \). Thus, we can factor the quadratic as follows:
\[
(x + 5)(x + 10) = 0
\]
Next, we set each factor equal to zero to find the values of \( x \):
1. \( x + 5 = 0 \)
\[
x = -5
\]
2. \( x + 10 = 0 \)
\[
x = -10
\]
Thus, the solutions for \( x \) are:
\[
\boxed{-5} \quad \text{and} \quad \boxed{-10}
\]
\[
x^2 + 21x + 50 - 6x = 0
\]
This simplifies to:
\[
x^2 + 15x + 50 = 0
\]
Next, we will factor the quadratic expression \( x^2 + 15x + 50 \). We need to find two numbers that multiply to 50 (the constant term) and add up to 15 (the coefficient of the linear term).
The pairs of factors of 50 are:
- \( 1 \times 50 \)
- \( 2 \times 25 \)
- \( 5 \times 10 \)
Among these, the pair \( 5 \) and \( 10 \) adds up to \( 15 \). Thus, we can factor the quadratic as follows:
\[
(x + 5)(x + 10) = 0
\]
Next, we set each factor equal to zero to find the values of \( x \):
1. \( x + 5 = 0 \)
\[
x = -5
\]
2. \( x + 10 = 0 \)
\[
x = -10
\]
Thus, the solutions for \( x \) are:
\[
\boxed{-5} \quad \text{and} \quad \boxed{-10}
\]