Question
A) Use factoring to find the solutions for 30 = − x
2 + 13x. (1 point)
x = 3, 10
x = − 10, − 3
x = − 3, 10
x = − 10, 3
2 + 13x. (1 point)
x = 3, 10
x = − 10, − 3
x = − 3, 10
x = − 10, 3
Answers
Answered by
GPT-4o mini
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 make factoring easier:
\[
x^2 - 13x + 30 = 0
\]
Now, we need to factor the quadratic expression \(x^2 - 13x + 30\). We are looking for two numbers that multiply to \(30\) and add to \(-13\). The two numbers that meet these criteria are \(-3\) and \(-10\), so we can factor the quadratic as follows:
\[
(x - 3)(x - 10) = 0
\]
Setting each factor to zero gives us the solutions:
1. \(x - 3 = 0 \quad \Rightarrow \quad x = 3\)
2. \(x - 10 = 0 \quad \Rightarrow \quad x = 10\)
Therefore, the solutions to the equation \(30 = -x^2 + 13x\) are:
\[
\boxed{x = 3, 10}
\]
\[
-x^2 + 13x - 30 = 0
\]
Next, we can multiply the entire equation by -1 to make factoring easier:
\[
x^2 - 13x + 30 = 0
\]
Now, we need to factor the quadratic expression \(x^2 - 13x + 30\). We are looking for two numbers that multiply to \(30\) and add to \(-13\). The two numbers that meet these criteria are \(-3\) and \(-10\), so we can factor the quadratic as follows:
\[
(x - 3)(x - 10) = 0
\]
Setting each factor to zero gives us the solutions:
1. \(x - 3 = 0 \quad \Rightarrow \quad x = 3\)
2. \(x - 10 = 0 \quad \Rightarrow \quad x = 10\)
Therefore, the solutions to the equation \(30 = -x^2 + 13x\) are:
\[
\boxed{x = 3, 10}
\]