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 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 3 comma 10
x=−3,10
Answers
GPT-4o mini
To solve the equation \(30 = -x^2 + 13x\) by factoring, we first rearrange it into 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 will factor the quadratic equation. We need two numbers that multiply to \(30\) (the constant term) and add up to \(-13\) (the coefficient of the linear term). The numbers \(-3\) and \(-10\) meet these criteria:
\[
(x - 3)(x - 10) = 0
\]
Setting each factor equal to zero gives us the solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 10 = 0 \quad \Rightarrow \quad x = 10
\]
Thus, the solutions are:
\[
x = 3, \quad x = 10
\]
So the correct response would be:
**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 will factor the quadratic equation. We need two numbers that multiply to \(30\) (the constant term) and add up to \(-13\) (the coefficient of the linear term). The numbers \(-3\) and \(-10\) meet these criteria:
\[
(x - 3)(x - 10) = 0
\]
Setting each factor equal to zero gives us the solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 10 = 0 \quad \Rightarrow \quad x = 10
\]
Thus, the solutions are:
\[
x = 3, \quad x = 10
\]
So the correct response would be:
**x = 3, 10**
**x equals 3 comma 10**