I know the guidelines state that we should post our working along with questions to enable people to direct us toward the correct solution but, I am more or less clueless with how I should take on this question. I will do my best to post anything of meaning or worth after the question, as usual any help is greatly appreciated with this. :)

Question
A firm's annual fixed cost of production is £7200. The annual variable costs of production are £60 per unit of output plis £2 per square unit of output. The product is sold at a constant price of £320 per unit. Calcualte the breakeven levels of production.

Ideas:

Breakeven occurs when Revenue = Costs.

Revenue = Selling Price * Items Sold so R(x) = 320 * x

Costs = Fixed set up costs + Variable costs * Items produced so C(x) = 7200+60+2 * x

B.E. = 320x - 7262x

I am not sure if this is the right direction to be going in or not and therefore can't / don't know how to progress any further.

Good news, I managed to reach the correct solution for this question. I'll post my working below for anyone that finds themselves in a similar situation!
If we allow x to be the number of outputs:
C(x) = 7200 + 60x + 2x^2 (Cost Function)
R(x) = 320x

P(x) = R - C
320x - 7200 + 60x + 2x^2
Once at this stage we can then rearrange the above stage into a quadratic equation : 2x^2 - 260x + 7200
Then solve using the following formulae: x = (-b+(OR)-√b^2-4ac) / 2a

a = to be = 260 c= 7200

After replacing these values you will end up with x= 40 and x= 90. Therefore the breakeven levels of production are 40 units and 90 units.

Yep, BE is when total revenue = total costs, or equivalently is the solution to

R(x) = C(x)

where x is the units of production. Total Revenue is straightforward at 320320x, but it looks like Total Cost should be

C(x) \ = \ 2x^2 \ + \ 60x \ + \ 7,200

(Note the difference in the question between "per unit of output" and "per square unit of output". I'm interpreting the latter as "per squared unit of output".)

R(x) = C(x) (breakeven) is equivalent to R(x) - C(x) = 0; which itself is just a symbolic expression of the fact that at breakeven, net income is zero.

Using your Revenue and Cost numbers in this "net income = zero" version of breakeven gives you

260x \ - \ 2x^2 \ - \ 7,200 \ = \ 0 \

The breakeven, in units x, is the solution to that quadratic, which you'll find by completing the square, or via the quadratic formula, e.g.. Hint: There are two solutions, both feasible since they're both positive, and hence there are actually two possible breakeven production levels.

Good luck!

You beat me to the punch, amigo... and you nailed it. Nice work!