Question

Solve the following quadratic equations by factorization:
(1/x-1)-(1/x-5)=6/7,x≠1,-5

Answer 1

$\alpha= \frac{6+\sqrt{123} }{3}$   and   $\beta = \frac{6-\sqrt{123} }{3}$

Step-by-step explanation:

$\frac{1}{X-1}-\frac{1}{X-5}=\frac{6}{7}$

$\frac{(X-5)-(X-1)}{(X-1)(X-5)}=\frac{6}{7}$

$\frac{X-5-X+1}{X^{2}-5X-X+5}=\frac{6}{7}$

$\frac{-4}{X^2-6X+5}=\frac{6}{7}$

$6x^2-36x+30=-28$

$6x^2-36+58=0$

$2(3x^2-18x-29)=0$

$3x^2-18x-29=0$

by using quadratic formula

a = 3,  b = -12 and  c = -29    

where  $D=b^2-4ac$

$D= (-12)^2-4\times3\times-29$

$D= 492$

$\alpha = \frac{-b+\sqrt{D} }{2a}$       and      $\beta = \frac{-b-\sqrt{D} }{2a}$

$\alpha= \frac{12+\sqrt{492} }{2\times3}$      and      $\beta = \frac{-12-\sqrt{492} }{2\times3}$  

$\alpha= \frac{6+\sqrt{123} }{3}$       and      $\beta = \frac{6-\sqrt{123} }{3}$