Question

Using quadratic formula solve (x-1) /(x-2) +(x-2) /(x-3) =4

Answer 1

$\huge{\pink{\underline {\ulcorner{\star\: Solution\: \star}\urcorner}}}$

$\frac{x-1}{x-2}+ \frac{x-2}{x-3} = 4\\\\ \frac{(x-1)(x-3)+(x-1)(x-2)}{(x-2)(x-3)} =4 \\\\ \frac{ x^{2} -3x-x+3 + x^{2} -2x-x+2}{ x^{2} -3x-2x+6}= 4\\\\ \frac{2x^{2} -4x-3x+5}{x^{2} -5x+6} =4\\\\ 2x^{2} -7x+5 = 4({x^{2} -5x+6})\\\\ 2x^{2} -7x+5 = 4x^{2} -20x +24\\\\ 0 = 4x^{2} -2x^{2} -20x+7x+24-5\\\\ 2x^{2} -13x+21=0$

Quadratic formula

a= 2

b= -13

c= 21

$\red{\bold{ Case\: - \: 1}}$

$x =\frac{-b + \sqrt{b^{2} -4ac}}{2a}\\\\\implies x= \frac{-(-13)+\sqrt{ (-13)^{2} - 4(2)(21)}}{2(2)}\\\\\implies x= \frac{ 13+\sqrt{169-168}}{4} \\\\\implies x= \frac{13+\sqrt{1}}{4}\\\implies x=\frac{13+1}{4}\\\implies x=3.5$

$\red{\bold{ Case\: - \: 2}}$

$x =\frac{-b - \sqrt{b^{2} -4ac}}{2a}\\\\\implies x= \frac{-(-13)-\sqrt{ (-13)^{2} - 4(2)(21)}}{2(2)}\\\\\implies x= \frac{ 13-\sqrt{169-168}}{4} \\\\\implies x= \frac{13-\sqrt{1}}{4}\\\implies x=\frac{13-1}{4}\\\implies x=3$

$\red{\bold{\boxed{ x=\: 3.5\: or \: 3}}}$