Question
Find all the solutions for x^2 - 2x + 49 = 11x + 7
Answers
Answered by
GPT-4o mini
To solve the equation \( x^2 - 2x + 49 = 11x + 7 \), we will first rearrange the equation so that all terms are on one side.
Start by moving all terms to the left side:
\[
x^2 - 2x + 49 - 11x - 7 = 0
\]
Combine like terms:
\[
x^2 - 2x - 11x + 49 - 7 = 0
\]
\[
x^2 - 13x + 42 = 0
\]
Next, we can factor the quadratic equation \( x^2 - 13x + 42 = 0 \). We are looking for two numbers that multiply to \( 42 \) and add up to \( -13 \).
The numbers \( -6 \) and \( -7 \) satisfy these conditions since:
\[
-6 \cdot -7 = 42 \quad \text{and} \quad -6 + (-7) = -13
\]
Thus, we can factor the equation as follows:
\[
(x - 6)(x - 7) = 0
\]
Setting each factor equal to zero gives us:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
Finally, the solutions to the equation \( x^2 - 2x + 49 = 11x + 7 \) are:
\[
\boxed{6} \quad \text{and} \quad \boxed{7}
\]
Start by moving all terms to the left side:
\[
x^2 - 2x + 49 - 11x - 7 = 0
\]
Combine like terms:
\[
x^2 - 2x - 11x + 49 - 7 = 0
\]
\[
x^2 - 13x + 42 = 0
\]
Next, we can factor the quadratic equation \( x^2 - 13x + 42 = 0 \). We are looking for two numbers that multiply to \( 42 \) and add up to \( -13 \).
The numbers \( -6 \) and \( -7 \) satisfy these conditions since:
\[
-6 \cdot -7 = 42 \quad \text{and} \quad -6 + (-7) = -13
\]
Thus, we can factor the equation as follows:
\[
(x - 6)(x - 7) = 0
\]
Setting each factor equal to zero gives us:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
Finally, the solutions to the equation \( x^2 - 2x + 49 = 11x + 7 \) are:
\[
\boxed{6} \quad \text{and} \quad \boxed{7}
\]