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Mathematics

Solve :

xa−a+bx=b(a+b)ax\dfrac{x}{a} - \dfrac{a + b}{x} = \dfrac{b(a + b)}{ax}, when x ≠ 0 and a ≠ 0.

Quadratic Equations

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Answer

Solving :

⇒xa−a+bx=b(a+b)ax⇒x2−a(a+b)ax=ab+b2ax⇒x2−a(a+b)ax×ax=ab+b2⇒x2−a2−ab=ab+b2⇒x2=a2+ab+ab+b2⇒x2=a2+2ab+b2⇒x2=(a+b)2⇒x=(a+b)2⇒x=(a+b),−(a+b).\Rightarrow \dfrac{x}{a} - \dfrac{a + b}{x} = \dfrac{b(a + b)}{ax} \\[1em] \Rightarrow \dfrac{x^2 - a(a + b)}{ax} = \dfrac{ab + b^2}{ax} \\[1em] \Rightarrow \dfrac{x^2 - a(a + b)}{ax} \times ax = ab + b^2 \\[1em] \Rightarrow x^2 - a^2 - ab = ab + b^2 \\[1em] \Rightarrow x^2 = a^2 + ab + ab + b^2 \\[1em] \Rightarrow x^2 = a^2 + 2ab + b^2 \\[1em] \Rightarrow x^2 = (a + b)^2 \\[1em] \Rightarrow x = \sqrt{(a + b)^2} \\[1em] \Rightarrow x = (a + b), -(a + b).

Hence, x = (a + b), -(a + b).

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