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Mathematics

Solve for x : 5x+(x2−1)5x−(x2−1)=75\dfrac{5x + (x^2 - 1)}{5x - (x^2 - 1)} = \dfrac{7}{5}.

Ratio Proportion

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Answer

Given,

5x+(x2−1)5x−(x2−1)=75\dfrac{5x + (x^2 - 1)}{5x - (x^2 - 1)} = \dfrac{7}{5}

Applying componendo and dividendo, we get :

⇒5x+(x2−1)+5x−(x2−1)5x+(x2−1)−[5x−(x2−1)]=7+57−5⇒10x5x−5x+(x2−1)+(x2−1)=122⇒10x2(x2−1)=6⇒5xx2−1=6⇒5x=6(x2−1)⇒5x=6x2−6⇒6x2−5x−6=0⇒6x2−9x+4x−6=0⇒3x(2x−3)+2(2x−3)=0⇒(3x+2)(2x−3)=0⇒3x+2=0 or 2x−3=0⇒3x=−2 or 2x=3⇒x=−23 or x=32.\Rightarrow \dfrac{5x + (x^2 - 1) + 5x - (x^2 - 1)}{5x + (x^2 - 1) - [5x - (x^2 - 1)]} = \dfrac{7 + 5}{7 - 5} \\[1em] \Rightarrow \dfrac{10x}{5x - 5x + (x^2 - 1) + (x^2 - 1)} = \dfrac{12}{2} \\[1em] \Rightarrow \dfrac{10x}{2(x^2 - 1)} = 6 \\[1em] \Rightarrow \dfrac{5x}{x^2 - 1} = 6 \\[1em] \Rightarrow 5x = 6(x^2 - 1) \\[1em] \Rightarrow 5x = 6x^2 - 6 \\[1em] \Rightarrow 6x^2 - 5x - 6 = 0 \\[1em] \Rightarrow 6x^2 - 9x + 4x - 6 = 0 \\[1em] \Rightarrow 3x(2x - 3) + 2(2x - 3) = 0 \\[1em] \Rightarrow (3x + 2)(2x - 3) = 0 \\[1em] \Rightarrow 3x + 2 = 0 \text{ or } 2x - 3 = 0 \\[1em] \Rightarrow 3x = -2 \text{ or } 2x = 3 \\[1em] \Rightarrow x = -\dfrac{2}{3} \text{ or } x = \dfrac{3}{2}.

Hence, x = −23 or 32-\dfrac{2}{3} \text{ or } \dfrac{3}{2}.

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