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Solve quadratic equation by factorization

x-a/ x-b +x -b/ x-a = a/b+ b/a

Posted by Sudhanshu Yadav 7 years, 3 months ago

CBSE > Class 10 > Mathematics

1 answers

Neeraj Sharma 7 years, 3 months ago

$\frac{x - a}{x - b} + \frac{x - b}{x - a} = \frac{a}{b} + \frac{b}{a}$

$l e t \frac{x - a}{x - b} = y \Rightarrow \frac{x - b}{x - a} = \frac{1}{y}$

$\Rightarrow y + \frac{1}{y} = \frac{a^{2} + b^{2}}{a b}$

$\Rightarrow \frac{y^{2} + 1}{y} = \frac{a^{2} + b^{2}}{a b} \Rightarrow a b y^{2} - a^{2} y - b^{2} y - a b = 0$

$\Rightarrow a y (b y - a) - b (b y - a) = 0 \Rightarrow (b y - a) (a y - b) = 0$

$\Rightarrow y = \frac{a}{b} o r y = \frac{b}{a}$

$c a s e - 1$

$i f y = \frac{a}{b} \Rightarrow \frac{x - a}{x - b} = \frac{a}{b} \Rightarrow b x - b a = a x - b a \Rightarrow b x = a x$

$b x - a x = 0 \Rightarrow x (b - a) = 0 \Rightarrow x = 0$

$C a s e - 2$

$i f y = \frac{b}{a} \Rightarrow \frac{x - a}{x - b} = \frac{b}{a} \Rightarrow a x - a^{2} = b x - b^{2}$

$\Rightarrow a x - b x = a^{2} - b^{2} \Rightarrow x (a - b) = (a - b) (a + b)$

$x = (a + b)$

$H e n c e x = 0 o r (a + b)$

0Thank You

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