Question

Use factoring to find the solutions for 30=−x2+13x
.(1 point)
Responses

x=3,10
x is equal to 3 comma 10 - no response given

x=−10,−3
x is equal to negative 10 comma negative 3 - incorrect

x=−10,3
x is equal to negative 10 comma 3 - no response given

x=−3,10

Answer 1

To solve the equation \(30 = -x^2 + 13x\) using factoring, we can start by rearranging 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 need to factor the quadratic equation \(x^2 - 13x + 30\). We are looking for two numbers that multiply to 30 (the constant term) and add up to -13 (the coefficient of \(x\)). The numbers that satisfy this condition are -3 and -10, since:

\[
-3 \times -10 = 30 \quad \text{and} \quad -3 + -10 = -13
\]

Thus, we can factor the quadratic as:

\[
(x - 3)(x - 10) = 0
\]

Setting each factor equal to zero gives:

1. \(x - 3 = 0\) ⟹ \(x = 3\)
2. \(x - 10 = 0\) ⟹ \(x = 10\)

Therefore, the solutions to the equation \(30 = -x^2 + 13x\) are:

\[
x = 3 \quad \text{and} \quad x = 10
\]

So the correct response is:

**x = 3, 10**