Question
n^2+3n-12=6 solve factoring
Answers
GPT-4o mini
To solve the equation \( n^2 + 3n - 12 = 6 \) by factoring, we first move all terms to one side to set the equation to zero:
\[
n^2 + 3n - 12 - 6 = 0
\]
This simplifies to:
\[
n^2 + 3n - 18 = 0
\]
Next, we need to factor the quadratic expression. We are looking for two numbers that multiply to \(-18\) (the constant term) and add up to \(3\) (the coefficient of the linear term).
The pairs of factors of \(-18\) that add up to \(3\) are \(6\) and \(-3\) (since \(6 \times -3 = -18\) and \(6 + (-3) = 3\)).
Thus, we can factor the quadratic as:
\[
(n + 6)(n - 3) = 0
\]
Now, we set each factor equal to zero:
1. \(n + 6 = 0\) ⟹ \(n = -6\)
2. \(n - 3 = 0\) ⟹ \(n = 3\)
So the solutions to the equation \(n^2 + 3n - 12 = 6\) are:
\[
n = -6 \quad \text{and} \quad n = 3
\]
\[
n^2 + 3n - 12 - 6 = 0
\]
This simplifies to:
\[
n^2 + 3n - 18 = 0
\]
Next, we need to factor the quadratic expression. We are looking for two numbers that multiply to \(-18\) (the constant term) and add up to \(3\) (the coefficient of the linear term).
The pairs of factors of \(-18\) that add up to \(3\) are \(6\) and \(-3\) (since \(6 \times -3 = -18\) and \(6 + (-3) = 3\)).
Thus, we can factor the quadratic as:
\[
(n + 6)(n - 3) = 0
\]
Now, we set each factor equal to zero:
1. \(n + 6 = 0\) ⟹ \(n = -6\)
2. \(n - 3 = 0\) ⟹ \(n = 3\)
So the solutions to the equation \(n^2 + 3n - 12 = 6\) are:
\[
n = -6 \quad \text{and} \quad n = 3
\]