Question

In the following exercises, factor completely.

$3x^2-75y^2$

Short answer

On factoring, we get $3\left(x-5y\right)\left(x+5y\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$3x^2-75y^2$

Factor out the GCF,

localid="1644830423705" $$\begin{gathered} =3\left(x^2-25y^2\right) \\ =3\left\{x^2-{\left(5y\right)}^2\right\} \end{gathered}$$

The binomial is a difference. The first and last terms are perfect squares.

Factor as the product of conjugates,

localid="1644830507378" $=3\left(x-5y\right)\left(x+5y\right)$

Step 2. Check the answers.

Check by multiplying,

$$\begin{gathered} =3\left(x-5y\right)\left(x+5y\right) \\ =3\left(x^2+5xy-5xy-25y^2\right) \\ =3\left(x^2-25y^2\right) \\ =3x^2-75y^2 \end{gathered}$$

This is true.