Question
Use factoring to find the solutions for 30=−x2+13x
.(1 point)
Responses
x=−10,−3
x equals negative 10 comma negative 3
x=−10,3
x equals negative 10 comma 3
x=−3,10
x equals negative 3 comma 10
x=3,10
.(1 point)
Responses
x=−10,−3
x equals negative 10 comma negative 3
x=−10,3
x equals negative 10 comma 3
x=−3,10
x equals negative 3 comma 10
x=3,10
Answers
GPT-4o mini
To solve the equation \( 30 = -x^2 + 13x \) by factoring, we first rewrite it in standard form:
\[
-x^2 + 13x - 30 = 0
\]
Multiplying through by -1 to make the leading coefficient positive, we get:
\[
x^2 - 13x + 30 = 0
\]
Next, we factor the quadratic expression. We need to find two numbers that multiply to \( 30 \) (the constant term) and add up to \( -13 \) (the coefficient of the \( x \) term). The pair of numbers that satisfy this is \( -3 \) and \( -10 \):
\[
(x - 3)(x - 10) = 0
\]
Now, we set each factor equal to zero:
1. \( x - 3 = 0 \) ⇒ \( x = 3 \)
2. \( x - 10 = 0 \) ⇒ \( x = 10 \)
Thus, the solutions to the equation are:
\[
x = 3, 10
\]
Therefore, the correct response is:
**x=3,10** (x equals 3 comma 10)
\[
-x^2 + 13x - 30 = 0
\]
Multiplying through by -1 to make the leading coefficient positive, we get:
\[
x^2 - 13x + 30 = 0
\]
Next, we factor the quadratic expression. We need to find two numbers that multiply to \( 30 \) (the constant term) and add up to \( -13 \) (the coefficient of the \( x \) term). The pair of numbers that satisfy this is \( -3 \) and \( -10 \):
\[
(x - 3)(x - 10) = 0
\]
Now, we set each factor equal to zero:
1. \( x - 3 = 0 \) ⇒ \( x = 3 \)
2. \( x - 10 = 0 \) ⇒ \( x = 10 \)
Thus, the solutions to the equation are:
\[
x = 3, 10
\]
Therefore, the correct response is:
**x=3,10** (x equals 3 comma 10)