Question
Solve for all values of xx by factoring.
x, squared, minus, 12, x, minus, 5, equals, minus, 6, x, plus, 2
x
2
−12x−5=
−6x+2
x, squared, minus, 12, x, minus, 5, equals, minus, 6, x, plus, 2
x
2
−12x−5=
−6x+2
Answers
GPT-4o mini
To solve the equation \(x^2 - 12x - 5 = -6x + 2\) by factoring, first, we should move all terms to one side of the equation:
\[
x^2 - 12x - 5 + 6x - 2 = 0
\]
Combine like terms:
\[
x^2 - 6x - 7 = 0
\]
Now we need to factor the quadratic equation \(x^2 - 6x - 7\). We are looking for two numbers that multiply to \(-7\) (the constant term) and add to \(-6\) (the coefficient of the \(x\) term). These numbers are \(-7\) and \(1\).
Thus, we can factor the equation as follows:
\[
(x - 7)(x + 1) = 0
\]
Now, we can set each factor equal to zero:
1. \(x - 7 = 0\)
2. \(x + 1 = 0\)
Solving these gives us:
1. \(x = 7\)
2. \(x = -1\)
Therefore, the solutions to the equation are:
\[
\boxed{7} \quad \text{and} \quad \boxed{-1}
\]
\[
x^2 - 12x - 5 + 6x - 2 = 0
\]
Combine like terms:
\[
x^2 - 6x - 7 = 0
\]
Now we need to factor the quadratic equation \(x^2 - 6x - 7\). We are looking for two numbers that multiply to \(-7\) (the constant term) and add to \(-6\) (the coefficient of the \(x\) term). These numbers are \(-7\) and \(1\).
Thus, we can factor the equation as follows:
\[
(x - 7)(x + 1) = 0
\]
Now, we can set each factor equal to zero:
1. \(x - 7 = 0\)
2. \(x + 1 = 0\)
Solving these gives us:
1. \(x = 7\)
2. \(x = -1\)
Therefore, the solutions to the equation are:
\[
\boxed{7} \quad \text{and} \quad \boxed{-1}
\]