For my Di, I have come up with a conjecture but through many attempts of proving it i am struggling and cannot prove it.

Question

So the expression is (r-2x)(s-2x)x, and because the middle term once expanded does not share either r or s, i could not go further with the conjecture. I need to find the derivative of the cubic equation and then use the quadratic formula to find an exact value of x. which should be the conjectured value of x: (r+s)±√(r²-rs+s² 

                                                                                                                                                        6

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 for clarification, all of x=(r+s)±√(r²-rs+s²  is divided by 6 as shown above. 

Answer 1

(r-2x)(s-2x)x=rsx-2x²(r+s)+4x³.

Derivative: rs-4x(r+s)+12x² or the quadratic:

12x²-4x(r+s)+rs.

The roots can be found using the formula:

x=(4(r+s)±√(16(r+s)²-48rs))/24.

Divide through by 4:

x=(r+s±√(r²+2rs+s²-3rs))/6,

x=(r+s±√(r²-rs+s²))/6.