Question

Factor completely.
6n^2 –30n+36

Answer 1

To factor the quadratic expression \(6n^2 - 30n + 36\), we can follow these steps:

1. **Factor out the greatest common factor (GCF)**:
The GCF of the coefficients \(6\), \(-30\), and \(36\) is \(6\). We factor \(6\) out of the expression:
\[
6(n^2 - 5n + 6)
\]

2. **Factor the quadratic \(n^2 - 5n + 6\)**:
Next, we will look for two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of \(n\)). The numbers \(-2\) and \(-3\) fit this requirement because:
\[
-2 \times -3 = 6 \quad \text{and} \quad -2 + (-3) = -5
\]
Thus, we can factor \(n^2 - 5n + 6\) as:
\[
(n - 2)(n - 3)
\]

3. **Combine the factors**:
Now, we can combine the factor we pulled out with the factors of the quadratic:
\[
6(n - 2)(n - 3)
\]

So, the complete factorization of the expression \(6n^2 - 30n + 36\) is:
\[
\boxed{6(n - 2)(n - 3)}
\]