Question
Which equation has the same solution as x, squared, minus, 13, x, plus, 12, equals, 10x
2
−13x+12=10?
Answer
Attempt 1 out of 3
The number of attempts on this specific problem before you are marked incorrect
Multiple Choice Answers
left parenthesis, x, minus, 6, point, 5, right parenthesis, squared, equals, 40, point, 2, 5(x−6.5)
2
=40.25
left parenthesis, x, plus, 6, point, 5, right parenthesis, squared, equals, 40, point, 2, 5(x+6.5)
2
=40.25
left parenthesis, x, minus, 6, point, 5, right parenthesis, squared, equals, minus, 44, point, 2, 5(x−6.5)
2
=−44.25
left parenthesis, x, plus, 6, point, 5, right parenthesis, squared, equals, minus, 44, point, 2, 5(x+6.5)
2
=−44.25
2
−13x+12=10?
Answer
Attempt 1 out of 3
The number of attempts on this specific problem before you are marked incorrect
Multiple Choice Answers
left parenthesis, x, minus, 6, point, 5, right parenthesis, squared, equals, 40, point, 2, 5(x−6.5)
2
=40.25
left parenthesis, x, plus, 6, point, 5, right parenthesis, squared, equals, 40, point, 2, 5(x+6.5)
2
=40.25
left parenthesis, x, minus, 6, point, 5, right parenthesis, squared, equals, minus, 44, point, 2, 5(x−6.5)
2
=−44.25
left parenthesis, x, plus, 6, point, 5, right parenthesis, squared, equals, minus, 44, point, 2, 5(x+6.5)
2
=−44.25
Answers
GPT-4o mini
To solve the equation \( x^2 - 13x + 12 = 10 \), we first simplify it:
1. Rewrite the equation as:
\[
x^2 - 13x + 12 - 10 = 0
\]
This simplifies to:
\[
x^2 - 13x + 2 = 0
\]
Now, we can calculate the roots of this quadratic equation or look for an equivalent equation by completing the square or using another technique.
### Completing the Square
To complete the square for \( x^2 - 13x + 2 \):
1. Take the coefficient of \( x \) (which is -13), divide it by 2, and square it:
\[
\left(\frac{-13}{2}\right)^2 = \frac{169}{4}
\]
2. Add and subtract this value to the equation:
\[
x^2 - 13x + \frac{169}{4} - \frac{169}{4} + 2 = 0
\]
3. Rewrite it:
\[
\left(x - \frac{13}{2}\right)^2 - \frac{169}{4} + 2 = 0
\]
Here, \( 2 = \frac{8}{4} \), so:
\[
\left(x - \frac{13}{2}\right)^2 - \frac{169 - 8}{4} = 0
\]
\[
\left(x - \frac{13}{2}\right)^2 - \frac{161}{4} = 0
\]
4. Rewrite it as:
\[
\left(x - \frac{13}{2}\right)^2 = \frac{161}{4}
\]
This isn't one of the choices you provided. However, let's look at what \( \frac{161}{4} \) simplifies to and if we can rewrite it differently.
Using an approximate decimal, \( \frac{161}{4} = 40.25 \).
Since none of the answers directly match, but we recognize that completing the square yields something similar, we can see that if we want:
\[
\left(x - \frac{13}{2}\right)^2 = 40.25
\]
could be similar to either of the first two choices, if we convert \( \frac{13}{2} \) to a decimal of \( 6.5 \):
Therefore, the equation:
\[
\left(x - 6.5\right)^2 = 40.25
\]
is a match for the first option given.
So, the correct answer from the multiple choice options is:
1. **\[ \left(x - 6.5\right)^2 = 40.25 \]**
1. Rewrite the equation as:
\[
x^2 - 13x + 12 - 10 = 0
\]
This simplifies to:
\[
x^2 - 13x + 2 = 0
\]
Now, we can calculate the roots of this quadratic equation or look for an equivalent equation by completing the square or using another technique.
### Completing the Square
To complete the square for \( x^2 - 13x + 2 \):
1. Take the coefficient of \( x \) (which is -13), divide it by 2, and square it:
\[
\left(\frac{-13}{2}\right)^2 = \frac{169}{4}
\]
2. Add and subtract this value to the equation:
\[
x^2 - 13x + \frac{169}{4} - \frac{169}{4} + 2 = 0
\]
3. Rewrite it:
\[
\left(x - \frac{13}{2}\right)^2 - \frac{169}{4} + 2 = 0
\]
Here, \( 2 = \frac{8}{4} \), so:
\[
\left(x - \frac{13}{2}\right)^2 - \frac{169 - 8}{4} = 0
\]
\[
\left(x - \frac{13}{2}\right)^2 - \frac{161}{4} = 0
\]
4. Rewrite it as:
\[
\left(x - \frac{13}{2}\right)^2 = \frac{161}{4}
\]
This isn't one of the choices you provided. However, let's look at what \( \frac{161}{4} \) simplifies to and if we can rewrite it differently.
Using an approximate decimal, \( \frac{161}{4} = 40.25 \).
Since none of the answers directly match, but we recognize that completing the square yields something similar, we can see that if we want:
\[
\left(x - \frac{13}{2}\right)^2 = 40.25
\]
could be similar to either of the first two choices, if we convert \( \frac{13}{2} \) to a decimal of \( 6.5 \):
Therefore, the equation:
\[
\left(x - 6.5\right)^2 = 40.25
\]
is a match for the first option given.
So, the correct answer from the multiple choice options is:
1. **\[ \left(x - 6.5\right)^2 = 40.25 \]**