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Mathematics

If x ≠ 0 and a ≠ 0, solve :

xa−a+bx=b(a+b)ax\dfrac{x}{a} - \dfrac{a + b}{x} = \dfrac{b(a + b)}{ax}

Quadratic Equations

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Answer

Given,

⇒xa−a+bx=b(a+b)ax⇒x2−a(a+b)ax=ab+b2ax⇒x2−a2−abax×ax=ab+b2⇒x2−a2−ab=ab+b2⇒x2=a2+ab+ab+b2⇒x2=a2+2ab+b2⇒x2=(a+b)2⇒x2−(a+b)2=0⇒(x−(a+b))(x+(a+b))=0⇒x=a+b or x=−(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^2 - ab}{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^2 - (a + b)^2 = 0 \\[1em] \Rightarrow (x - (a + b))(x + (a + b)) = 0 \\[1em] \Rightarrow x = a + b \text{ or } x = -(a + b).

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

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