Question

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

x=−3,10
x equals negative 3 comma 10

x=−10,−3
x equals negative 10 comma negative 3

x=3,10
x equals 3 comma 10

x=−10,3

Answer 1

To solve the equation \(30 = -x^2 + 13x\), we can first rearrange it to standard quadratic form:

\[
-x^2 + 13x - 30 = 0
\]

Multiplying the entire equation by -1 to make the coefficient of \(x^2\) positive gives us:

\[
x^2 - 13x + 30 = 0
\]

Next, we can factor the quadratic equation. 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 fit these criteria are \(3\) and \(10\) because:

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

So we can write the factored form as:

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

Now, we set each factor equal to zero to find the solutions:

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

Thus, the solutions to the equation are:

\[
x = 3, 10
\]

The correct response is:

**x equals 3, 10**.