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 negative 3
x=−10,3
x equals negative 10 comma 3
x=−3,10
.(1 point)
Responses
x=3,10
x equals 3 comma 10
x=−10,−3
x equals negative 10 comma negative 3
x=−10,3
x equals negative 10 comma 3
x=−3,10
Answers
GPT-4o mini
To solve the equation \( 30 = -x^2 + 13x \) using factoring, we first rearrange it into standard form:
\[
-x^2 + 13x - 30 = 0
\]
Now, we can multiply the entire equation by -1 to make the leading coefficient positive:
\[
x^2 - 13x + 30 = 0
\]
Next, 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 these conditions are \( 3 \) and \( 10 \) (since \( 3 \times 10 = 30 \) and \( 3 + 10 = 13 \)). Thus, we can rewrite the equation as:
\[
(x - 3)(x - 10) = 0
\]
To find the solutions, we set each factor to zero:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 10 = 0 \) → \( x = 10 \)
Therefore, the solutions to the equation are:
\[
x = 3, 10
\]
So the correct response is:
**x=3,10**
\[
-x^2 + 13x - 30 = 0
\]
Now, we can multiply the entire equation by -1 to make the leading coefficient positive:
\[
x^2 - 13x + 30 = 0
\]
Next, 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 these conditions are \( 3 \) and \( 10 \) (since \( 3 \times 10 = 30 \) and \( 3 + 10 = 13 \)). Thus, we can rewrite the equation as:
\[
(x - 3)(x - 10) = 0
\]
To find the solutions, we set each factor to zero:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 10 = 0 \) → \( x = 10 \)
Therefore, the solutions to the equation are:
\[
x = 3, 10
\]
So the correct response is:
**x=3,10**