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Mathematics

Using the properties of proportion, solve the following equation for x;

given x3+3x3x2+1=34191.\dfrac{x^3 + 3x}{3x^2 + 1} = \dfrac{341}{91}.

Ratio Proportion

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Answer

Given,

x3+3x3x2+1=34191\dfrac{x^3 + 3x}{3x^2 + 1} = \dfrac{341}{91}

Applying componendo and dividendo,

⇒x3+3x+3x2+1x3+3x−3x2−1=341+91341−91⇒(x+1)3(x−1)3=432250⇒(x+1)3(x−1)3=216125⇒(x+1x−1)3=(65)3⇒x+1x−1=65\Rightarrow \dfrac{x^3 + 3x + 3x^2 + 1}{x^3 + 3x - 3x^2 - 1} = \dfrac{341 + 91}{341 - 91} \\[1em] \Rightarrow \dfrac{(x + 1)^3}{(x - 1)^3} = \dfrac{432}{250} \\[1em] \Rightarrow \dfrac{(x + 1)^3}{(x - 1)^3} = \dfrac{216}{125} \\[1em] \Rightarrow \Big(\dfrac{x + 1}{x - 1}\Big)^3 = \Big(\dfrac{6}{5}\Big)^3 \\[1em] \Rightarrow \dfrac{x + 1}{x - 1} = \dfrac{6}{5}

Again applying componendo and dividendo,

⇒x+1+x−1x+1−x+1=6+56−5⇒2x2=111⇒x=11.\Rightarrow \dfrac{x + \cancel{1} + x - \cancel{1}}{\cancel{x} + 1 - \cancel{x} + 1} = \dfrac{6 + 5}{6 - 5} \\[1em] \Rightarrow \dfrac{2x}{2} = \dfrac{11}{1} \\[1em] \Rightarrow x = 11.

Hence, the value of x is 11.

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