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