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So in order to derive the quadratic equation we need to start with a quadratic that can represent any quadratic equation. Lets start with ax²+bx+c=0.

We will use the completing the squar...

each share is 6000 / 12 - Imran's share is 7x this = 3500 rupees. 

3500 x 3 / 5 = 2100. 

2100/6000 x 100 = 35%

To solve quadratic equations we first need to factorise the equation into brackets. We can do this by looking at the numbers in the quadratic equation. The only number that isn't an x coefficient is the 1...

So we have y = 0 and therefore x² - 6x - 16 =0 

This is then factorised to (x+2)(x-8) = 0 and therefore we have x = -2, x = -8

To find the gradient we need to find dy/dx

So...

For the most part, this comes down to practice. The insight in seeing which identity gives you the most efficient path to the solution comes with experience, however a little trick I like to use (silly as...