Question

a/x-b + b/ x-a = 2 , x does not equal to a,b ​

Answer 1

Given Question :-

$\tt \: Solve \: for \: x : \: \dfrac{a}{x - b} + \dfrac{b}{x - a} = 2$

$\large\underline{\bold{Solution-}}$

$\rm :\longmapsto\:\dfrac{a}{x - b} + \dfrac{b}{x - a} = 2$

$\rm :\longmapsto\:\dfrac{a}{x - b} + \dfrac{b}{x - a} = 1 + 1$

$\rm :\longmapsto\:\dfrac{a}{x - b} - 1+ \dfrac{b}{x - a} - 1 = 0$

$\rm :\longmapsto\:\bigg(\dfrac{a - (x - b)}{x - b} \bigg) + \bigg(\dfrac{b - (x - a)}{x - a} \bigg) = 0$

$\rm :\longmapsto\:\bigg(\dfrac{a - x + b}{x - b} \bigg) + \bigg(\dfrac{b - x + a}{x - a} \bigg) = 0$

$\rm :\longmapsto\:(a + b - x)\bigg( \dfrac{1}{x - b} + \dfrac{1}{x - a} \bigg) = 0$

$\rm :\longmapsto\:(a + b - x)\bigg( \dfrac{x - a + x - b}{(x - a)(x - b)} \bigg) = 0$

$\rm :\longmapsto\:(a + b - x)\bigg( \dfrac{2x - a - b}{(x - a)(x - b)} \bigg) = 0$

$\bf\implies \:x = a + b \: \: \: \: or \: x = \dfrac{a + b}{2}$

$\large\underline{\bold{Alternative \: Method-}}$

$\rm :\longmapsto\:\dfrac{a}{x - b} + \dfrac{b}{x - a} = 2$

$\rm :\longmapsto\:\dfrac{a(x - a) + b(x - b)}{(x - b)(x - a)} = 2$

$\rm :\longmapsto\:\dfrac{ax - {a}^{2} + bx - {b}^{2} }{ {x}^{2} - ax - bx + ab} = 2$

$\rm :\longmapsto\:ax - {a}^{2} - {b}^{2} + bx = 2 {x}^{2} - 2ax - 2bx + 2ab$

$\rm :\longmapsto\: {2x}^{2} - 3ax - 3bx + {a}^{2} + {b}^{2} + 2ab = 0$

$\rm :\longmapsto\: {2x}^{2} - 3x(a + b) + {(a + b)}^{2}$

$\rm :\longmapsto\: {2x}^{2} - 2x(a + b) - x(a + b) + {(a + b)}^{2}$

$\rm :\longmapsto\:2x(x - a - b) - (a + b)(x - a - b) = 0$

$\rm :\longmapsto\:(x - a - b)(2x - a - b) = 0$

$\bf\implies \:x = a + b \: \: \: \: or \: x = \dfrac{a + b}{2}$

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• There are various methods by which you can solve a quadratic equation.

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• By plotting graph.

Question : What are the forms of quadratic function?

• The three forms are as follow :

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