Question

eliminate x from the following​

Answer 1

Answer :

The required equation is 4p² - 4q² - 4q - 5 = 0

Step-by-step explanation :

Given :

x + 1/x = 2p

x - 1/x = 2q + 1

To find :

eliminate x from the given equations

Formulae :

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

Solution :

First equation :

x + 1/x = 2p

Squaring on both sides,

 $\sf (x+\dfrac{1}{x})^2=(2p)^2 \\\\ \sf x^2+(\dfrac{1}{x})^2+2(x)(\dfrac{1}{x})=4p^2 \\\\ \sf x^2+\dfrac{1}{x^2}+2=4p^2 \\\\ \sf x^2+\dfrac{1}{x^2}=4p^2-2 \longrightarrow [1]$

Second equation :

x - 1/x = 2q + 1

Squaring on both sides.

 $\sf (x-\dfrac{1}{x})^2=(2q+1)^2 \\\\ \sf x^2+(\dfrac{1}{x})^2-2(x)(\dfrac{1}{x})=(2q)^2+1^2+2(2q)(1) \\\\ \sf x^2+\dfrac{1}{x^2}-2=4q^2+1+4q \\\\ \sf x^2+\dfrac{1}{x^2}=4q^2+4q+1+2 \\\\ \sf x^2+\dfrac{1}{x^2}=4q^2+4q+3 \longrightarrow [2]$

x² + 1/x² = 4p² - 2 = 4q² + 4q + 3

 4p² - 2 = 4q² + 4q + 3

  4p² - 4q² = 4q + 3 + 2

   4p² - 4q² = 4q + 5

   4p² - 4q² - 4q - 5 = 0