Factorising (a-b)^2

To factorise the expression 3-12(a-b)^2, let's first understand the concept of factorising a perfect square trinomial.

A perfect square trinomial is an algebraic expression in the form (a-b)^2, where 'a' and 'b' represent any real numbers or variables. When we expand (a-b)^2, we get a^2 - 2ab + b^2.

Now, let's factorise the given expression step by step.

Step 1: Recognize the perfect square trinomial

In the given expression, (a-b)^2 is a perfect square trinomial. We can identify this because it follows the form of (a-b)^2, where 'a' is the term before the square and 'b' is the term after the square.

Step 2: Substitute the perfect square trinomial

Replace the perfect square trinomial (a-b)^2 with its expanded form: a^2 - 2ab + b^2.

So, the expression 3-12(a-b)^2 becomes 3 - 12(a^2 - 2ab + b^2).

Step 3: Distribute the factor

Distribute the factor 12 to each term inside the bracket: 3 - 12a^2 + 24ab - 12b^2.

Step 4: Combine like terms

Combine the like terms: -12a^2 + 24ab - 12b^2.

Step 5: Factorise the expression

Now, we can factorise the expression -12a^2 + 24ab - 12b^2. To do this, we need to find common factors among the terms.

Step 6: Find common factors

In this case, the common factor among the terms is -12. Factoring out -12, we get:

-12(a^2 - 2ab + b^2).

Step 7: Recognize the perfect square trinomial again

Now, we can see that (a^2 - 2ab + b^2) is a perfect square trinomial. It can be factored as (a-b)^2.

Final Step: Rewrite the expression in factored form

Substituting (a-b)^2 for (a^2 - 2ab + b^2), we have:

-12(a-b)^2.

Therefore, the given expression 3-12(a-b)^2 can be factorised as -12(a-b)^2.

Conclusion:

The expression 3-12(a-b)^2 can be factorised as -12(a-b)^2.