Hi,

Could you please show me a step by step worked solution to this question?

Factorise:

6p^2 - 17pq + 12pq^2

I am getting confused as what to do when there are two "squares".

the answer should be "(2p- 3q)(3p -4q)"

Thank you very much!!!

forget the q for the time being, add it in at the end

6 p^2 -17 p + 12
(2 p - 3)(3 p - 4)
now put the q back
(2 p - 3 q)(3 p - 4 q)

Sure! I can help you with that.

To factorise the expression 6p^2 - 17pq + 12pq^2, we need to look for common factors and try to express the expression as a product of two binomial expressions.

Step 1: Identify any common factors.
In this case, there are no common factors among all the terms.

Step 2: Look for patterns.
Notice that the expression is a quadratic trinomial, meaning it has three terms with two different variables, p and q. We can try to factorise it using the "ac method" or "trial and error" method.

Step 3: Factorise the expression using the "ac method".
The "ac method" involves finding two numbers that multiply to give the product of the coefficient of the squared term (6) and the constant term (12), and also add up to give the coefficient of the middle term (-17).

For the given expression, the coefficient of the squared term is 6 and the constant term is 12. The product of these is 72. We need to find two numbers that multiply to give 72 and add up to -17. The numbers -8 and -9 satisfy these conditions.

Step 4: Split the middle term using the two numbers identified above.
Rewrite the expression by splitting the middle term (-17pq) using -8pq and -9pq:
6p^2 - 8pq - 9pq + 12pq^2

Step 5: Group the terms.
Group the first two terms and the last two terms:
(6p^2 - 8pq) + (-9pq + 12pq^2)

Step 6: Factor out the greatest common factor from each group.
In the first group, the greatest common factor is 2p:
2p(3p - 4q)
In the second group, the greatest common factor is -3q:
-3q(3p - 4q)

Step 7: Rearrange the terms.
Combine the two factorised groups:
2p(3p - 4q) - 3q(3p - 4q)

Step 8: Factor out the common binomial factor.
Now we have a common binomial factor, (3p - 4q):
(2p - 3q)(3p - 4q)

So, the fully factorised form of 6p^2 - 17pq + 12pq^2 is (2p - 3q)(3p - 4q).

I hope this step-by-step explanation helps you understand how to factorise expressions with two squares. Let me know if you have any further questions!