Question
Solve for all values of xx by factoring.
x, squared, minus, 9, x, minus, 20, equals, minus, 2, x, minus, 2
x
2
−9x−20=
−2x−2
x, squared, minus, 9, x, minus, 20, equals, minus, 2, x, minus, 2
x
2
−9x−20=
−2x−2
Answers
GPT-4o mini
To solve the equation
\[
x^2 - 9x - 20 = -2x - 2,
\]
we first rearrange all terms to one side of the equation:
\[
x^2 - 9x + 2x - 20 + 2 = 0.
\]
Combining like terms, we have:
\[
x^2 - 7x - 18 = 0.
\]
Next, we need to factor the quadratic equation \(x^2 - 7x - 18\). We are looking for two numbers that multiply to \(-18\) (the constant term) and add up to \(-7\) (the coefficient of \(x\)).
The two numbers that satisfy these conditions are \(-9\) and \(2\), since:
\[
-9 \cdot 2 = -18 \quad \text{and} \quad -9 + 2 = -7.
\]
Thus, we can factor the expression as:
\[
(x - 9)(x + 2) = 0.
\]
Setting each factor equal to zero gives us the solutions:
1. \(x - 9 = 0\) leads to \(x = 9\),
2. \(x + 2 = 0\) leads to \(x = -2\).
So, the solutions to the equation \(x^2 - 9x - 20 = -2x - 2\) are:
\[
\boxed{9 \text{ and } -2}.
\]
\[
x^2 - 9x - 20 = -2x - 2,
\]
we first rearrange all terms to one side of the equation:
\[
x^2 - 9x + 2x - 20 + 2 = 0.
\]
Combining like terms, we have:
\[
x^2 - 7x - 18 = 0.
\]
Next, we need to factor the quadratic equation \(x^2 - 7x - 18\). We are looking for two numbers that multiply to \(-18\) (the constant term) and add up to \(-7\) (the coefficient of \(x\)).
The two numbers that satisfy these conditions are \(-9\) and \(2\), since:
\[
-9 \cdot 2 = -18 \quad \text{and} \quad -9 + 2 = -7.
\]
Thus, we can factor the expression as:
\[
(x - 9)(x + 2) = 0.
\]
Setting each factor equal to zero gives us the solutions:
1. \(x - 9 = 0\) leads to \(x = 9\),
2. \(x + 2 = 0\) leads to \(x = -2\).
So, the solutions to the equation \(x^2 - 9x - 20 = -2x - 2\) are:
\[
\boxed{9 \text{ and } -2}.
\]