Question

How do you factor the expression `12x^2 - 75`?

Answer 1

`12x^2 - 75 = color(green)(3(2x + 5) (2x - 5)`

Explanation:

Looking at this expression, we should get an idea that the factorization is probably based on the `color(blue)(a^2 - b^2 = (a+b)(a-b)` identity

But 12 and 75 are not perfect squares!

In such cases, we will almost always have a common factor among the two terms.

In the given expression, 3 is a common factor to both the terms.

`12x^2 - 75 = 3(4x^2 - 25)`

Now look at the expression within the brackets! 4 and 25 are perfect squares and the identity `color(blue)(a^2 - b^2 = (a+b)(a-b)` can be applied. It can be written as:

`3 times {(2x)^2 - 5^2}`

Applying the identity, we get:

`= color(green)(3(2x + 5) (2x - 5)`

The above expression cannot be factorised further and is hence the factorised form of the expression `12x^2 - 75`