Have i done this simultaneous equation correctly

G = 4F2 (SQUARED) -3
G+ 11F = 0

(4F2 - 3) + 11F =0

4F2 - 3 + 11F = 0

I DONT KNOW HOW TO FACTORISE THIS, IS IT POSSIBLE? AM I WRONG?

"I DONT KNOW HOW TO FACTORISE THIS"

You can always use the quadratic formula, just remember that the roots aren't necessarily REAL numbers if you use it.

Anyhoo, this one doesn't need the QF, you can factorise it by-eye:

4F^2 + 11F -3 = 0

Take the factors of 3, which are 1 & 3.

You know that quadratics always factorise into the form:

(aX+b)(cX+d)

b, d must be 1,-3 or -1,3 so as to make the -3 in the expanded result.

Let's try -1,3:

(aX-1)(cX+3)

Expand this back out and you can see:

acX^2 -cX +3aX -3 = 0 if (aX-1)(cX+3) represents your original quadratic.

If you let X=F, then ac = 4, 3a-c = 11

So, by observation you could see that if we let a=4, c=1 then ac=4 and 3a-c =11

So you have your factorised quadratic:

(4F-1)(F+3)=0

G = 4F^2 - 3

G + 11F = 0
G = -11F

4F^2 - 3 = -11F
4F^2 + 11F - 3 = 0
(4F - 1)(F + 3) = 0
F = {-3, 1/4}

or use the quadratic formula
(-11 ± √169) / 8
(-11 ± 13) / 8

F = {-3, 1/4}

or
4(x + 3)(x - .25) = 0

You're right!

4*F^2 + 11*F - 3 = (F + 3)*(4*F - 1) = 0

F + 3 = 0

F1 = - 3

4*F - 1 = 0

F2 = 1/4

G1 = - 11*( - 3) = 33

G2 = - 11*(1/4) = - 11/4