Question

3m^2-9m-30=0. Factor

Answer 1

To factor the quadratic equation \(3m^2 - 9m - 30 = 0\), we can start by factoring out the greatest common factor (GCF) from all terms.

The GCF of \(3m^2\), \(-9m\), and \(-30\) is 3.

First, we factor out 3:

\[
3(m^2 - 3m - 10) = 0
\]

Next, we need to factor the quadratic expression \(m^2 - 3m - 10\). We are looking for two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(m\)).

The numbers that fulfill this are \(-5\) and \(2\) because:

\[
-5 \times 2 = -10 \quad \text{and} \quad -5 + 2 = -3
\]

Thus, we can factor the quadratic as follows:

\[
m^2 - 3m - 10 = (m - 5)(m + 2)
\]

Putting it all together, the equation can now be expressed as:

\[
3(m - 5)(m + 2) = 0
\]

To solve for \(m\), we set each factor equal to zero:

1. \(m - 5 = 0 \Rightarrow m = 5\)
2. \(m + 2 = 0 \Rightarrow m = -2\)

The complete factorization of the original equation is:

\[
3(m - 5)(m + 2) = 0
\]

And the solutions to the equation \(3m^2 - 9m - 30 = 0\) are:

\[
m = 5 \quad \text{and} \quad m = -2
\]