Question

$\frac{5}{x - 6} - \frac{4}{x - 4} = \frac{3}{x + 2}$
​

Answer 1

ANSWER:

Given:

$\:\:\:\:\bullet\:\:\:\:\dfrac{5}{x-6}-\dfrac{4}{x-4}=\dfrac{3}{x+2}$

To Find:

$\:\:\:\:\bullet\:\:\:\:\text{Value of x}$

Solution:

$:\longrightarrow\dfrac{5}{x-6}-\dfrac{4}{x-4}=\dfrac{3}{x+2}\\\\\text{Taking LCM,}\\\\:\implies\dfrac{5(x-4)-4(x-6)}{(x-6)\times(x-4)}=\dfrac{3}{x+2}\\\\:\implies\dfrac{5x-20-4x+24}{(x^2-4x-6x+24)}=\dfrac{3}{x+2}\\\\:\implies\dfrac{x+4}{x^2-10x+24}=\dfrac{3}{x+2}\\\\\text{On cross-multiplying,}\\\\:\implies (x+4)\times(x+2)=3\times(x^2-10x+24)\\\\:\implies x^2+4x+2x+8=3x^2-30x+72\\\\:\implies x^2+6x+8=3x^2-30x+72\\\\:\implies (3x^2-30x+72)-(x^2+6x+8)=0$

$\\\\:\implies3x^2-30x+72-x^2-6x-8=0\\\\:\implies3x^2-x^2-30x-6x+72-8=0\\\\:\implies2x^2-36x+64=0\\\\:\implies2(x^2-18x+32)=0\\\\:\implies x^2-18x+32=0$

$\\\\\text{\sf{\underline{METHOD1: Middle Term Splitting:}}}\\\\:\implies x^2-18x+32=0\\\\:\implies x^2-16x-2x+32=0\\\\:\implies x(x-16)-2(x-16)=0\\\\:\implies (x-16)(x-2)=0\\\\\bf{:\implies\underline{x=16\:\:\:or\:\:\:2}}$

$\\\\\text{\sf{\underline{METHOD2: Quadratic Formula:}}}\\\\:\longrightarrow x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\text{In this, a=coefficient of x ^2 , b=coefficient of x, c=constant}\\\\\text{So, a=1, b=-18 and c=32}\\\\:\implies x=\dfrac{-(-18)\pm\sqrt{(-18)^2-4(1)(32)}}{2(1)}\\\\:\implies x=\dfrac{18\pm\sqrt{324-128}}{2}\\\\:\implies x=\dfrac{18\pm\sqrt{196}}{2}\\\\:\implies x=\dfrac{18\pm14}{2}\\\\:\implies x=9\pm7\\\\:\implies x=9+7 \:\:\:or\:\:\:x=9-7\\\\\bf{:\implies\underline{x=16\:\:\:or\:\:\:2}}$

Formula Used:

$\:\:\:\:\bullet\:\:\:\:\text{Quadratic Equation:\:\:\:}x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

Verification:

$:\longrightarrow\dfrac{5}{x-6}-\dfrac{4}{x-4}=\dfrac{3}{x+2}\\\\\text{For x=16}\\\\:\implies\dfrac{5}{16-6}-\dfrac{4}{16-4}=\dfrac{3}{16+2}\\\\:\implies\dfrac{5}{10}-\dfrac{4}{12}=\dfrac{3}{18}\\\\:\implies\dfrac{1}{2}-\dfrac{1}{3}=\dfrac{1}{6}\\\\:\implies\dfrac{3-2}{6}=\dfrac{1}{6}\\\\:\implies\dfrac{1}{6}=\dfrac{1}{6}- - - - - - - -(1)$

$\text{For x=2}\\\\:\implies \dfrac{5}{2-6}-\dfrac{4}{2-4}=\dfrac{3}{2+2}\\\\:\implies\dfrac{5}{-4}-\dfrac{4}{-2}=\dfrac{3}{4}\\\\:\implies\dfrac{-5}{4}-\dfrac{-2}{1}=\dfrac{3}{4}\\\\\:\implies\dfrac{-5}{4}+\dfrac{2}{1}=\dfrac{3}{4}\\\\:\implies\dfrac{-5+8}{4}=\dfrac{3}{4}\\\\:\implies\dfrac{3}{4}=\dfrac{3}{4}- - - - - - - -(2)\\\\\text{In (1) \& (2):}\\\\\text{As, LHS=RHS.}\\\\\text{\underline{HENCE VERIFIED}}$

Answer 2

Step-by-step explanation:

$\frac{5}{x - 6} - \frac{4}{x - 4} = \frac{3}{x + 2}$