Question

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

x=3,10
x equals 3 comma 10

x=−3,10
x equals negative 3 comma 10

x=−10,3
x equals negative 10 comma 3

x=−10,−3

Answer 1

To solve the equation \(30 = -x^2 + 13x\) by factoring, we first rearrange the equation to set it to zero:

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

Next, we can multiply the entire equation by -1 to simplify:

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

Now we need to factor the quadratic expression \(x^2 - 13x + 30\). We are looking for two numbers that multiply to 30 (the constant term) and add up to -13 (the coefficient of the x term).

The numbers that meet these criteria are -3 and -10, because:

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

This allows us to factor the expression as follows:

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

Setting each factor equal to zero gives us:

\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 10 = 0 \quad \Rightarrow \quad x = 10
\]

Thus, the solutions to the equation are:

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

In terms of the response options, the correct answer is:

**x=3,10**

So the complete answer is:

**x equals 3 comma 10**