Question

find the roots of following equation by factorization x+1/x-1 - x-1/x+1 = 5/6​

Answer 1

Answer:

5 and -0.2 are the roots of the given equation

Step-by-step explanation:

$\sf \dfrac{x+1}{x-1} - \dfrac{x-1}{x+1}=\dfrac{5}{6} \\\\\\ \sf \dfrac{(x+1)(x+1)}{(x-1)(x+1)} - \dfrac{(x-1)(x-1)}{(x+1)(x-1)}=\dfrac{5}{6} \\\\\\ \sf\dfrac{(x+1)^2}{(x-1)(x+1)} - \dfrac{(x-1)^2}{(x+1)(x-1)}=\dfrac{5}{6} \\\\\\ \sf \dfrac{(x+1)^2-[(x-1)^2]}{(x-1)(x+1)} =\dfrac{5}{6} \\\\\\ \sf \dfrac{x^2+1^2+2(x)(1)-[x^2+1^2-2(x)(1)]}{(x-1)(x+1)} =\dfrac{5}{6} \\\\\\ \sf \dfrac{x^2+1+2x-[x^2+1-2x]}{(x-1)(x+1)} =\dfrac{5}{6} \\\\\\ \dfrac{x^2+1+2x-x^2-1+2x}{(x-1)(x+1)} =\dfrac{5}{6}$

$\sf \dfrac{4x}{x(x+1)-1(x+1)} =\dfrac{5}{6} \\\\ \sf \dfrac{4x}{x^2+x-x-1} =\dfrac{5}{6} \\\\ \sf \dfrac{4}{x^2-1} =\dfrac{5}{6} \\\\ \sf 4x \times 6=5 \times (x^2-1) \\\\ \sf 24x=5x^2-5 \\\\ \sf 5x^2-24x-5=0$

The quadratic equation is 5x² - 24x - 5 = 0

Now, we've to factorize to find the roots.

5x² - 24x - 5 = 0

5x² - 25x + x - 5 = 0

5x(x - 5) + 1(x - 5) = 0

(x - 5) (5x + 1) = 0

[1] x - 5 = 0 ; x = +5

[2] 5x + 1 = 0 ; x = -1/5 = -0.2

Verification :

• Put x = 5,

$\sf \dfrac{x+1}{x-1} - \dfrac{x-1}{x+1}=\dfrac{5}{6} \\\\\\ \sf \dfrac{5+1}{5-1} - \dfrac{5-1}{5+1}=\dfrac{5}{6} \\\\ \sf \dfrac{6}{4} -\dfrac{4}{6}=\dfrac{5}{6} \\\\ \sf \dfrac{3}{2} -\dfrac{2}{3}=\dfrac{5}{6} \\\\ \sf \dfrac{3 \times 3}{2 \times 3} -\dfrac{2 \times 2}{3 \times 2}=\dfrac{5}{6} \\\\ \sf \dfrac{9}{6} -\dfrac{4}{6}=\dfrac{5}{6} \\\\ \sf \dfrac{9-4}{6} =\dfrac{5}{6} \\\\ \sf \dfrac{5}{6} =\dfrac{5}{6} \\\\ \sf LHS=RHS$

• Put x = -0.2

$\sf \dfrac{x+1}{x-1}-\dfrac{x-1}{x+1}=\dfrac{5}{6} \\\\\\ \sf\dfrac{-0.2+1}{-0.2-1}-\dfrac{-0.2-1}{-0.2+1}=\dfrac{5}{6} \\\\ \sf \dfrac{0.8}{-1.2}-\dfrac{-1.2}{0.8}=\dfrac{5}{6}\\\\ \sf\dfrac{8}{-12}-\dfrac{-12}{8}=\dfrac{5}{6} \\\\ \sf \dfrac{8}{-12}+\dfrac{12}{8}=\dfrac{5}{6}\\\\ \sf\dfrac{-2}{3} +\dfrac{3}{2}=\dfrac{5}{6} \\\\ \sf \dfrac{3}{2}-\dfrac{2}{3} =\dfrac{5}{6} \\\\ \sf \dfrac{9}{6}-\dfrac{4}{6}=\dfrac{5}{6}\\\\ \sf \dfrac{9-4}{6}=\dfrac{5}{6}\\\\ \sf \dfrac{5}{6} =\dfrac{5}{6} \\\\ \sf LHS=RHS$

Hence verified!