Question

solve this question and sorry for the handwriting

Answer 1

we can solve this question by using componendo and dividendo.

x+1/x-1 = √p+q + √p-q + √p+q - √p-q/
√p+q + √p-q - √p+q + √p-q

x+1/x-1 = 2√p+q/2√p-q
Then cancelling both 2.

x+1/x-1 = √p+q/√p-q

Squaring both sides
we get,

x^2 + 1 + 2x/x^2 + 1 - 2x = p+q/p-q

Again using componendo and dividendo we get,

2x^2 + 2/4x = 2p/2q
Taking 2 common on Left hand side and cancelling 2 on right hand side we get,

x^2 + 1/2x = p/q

Then cross multiplying

qx^2 + q = 2px
qx^2 - 2px + q = 0 -Hence proved.