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Mathematics

Using the properties of proportion, solve for x,

given x4+12x2=178\dfrac{x^4 + 1}{2x^2} = \dfrac{17}{8}

Ratio Proportion

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Answer

Given,

⇒x4+12x2=178\Rightarrow \dfrac{x^4 + 1}{2x^2} = \dfrac{17}{8}

Applying componendo and dividendo, we get

⇒x4+1+2x2x4+1−2x2=17+817−8⇒(x2+1)2(x2−1)2=259⇒x2+1x2−1=±53\Rightarrow \dfrac{x^4 + 1 + 2x^2}{x^4 + 1 - 2x^2} = \dfrac{17 + 8}{17 - 8} \\[1em] \Rightarrow \dfrac{(x^2 + 1)^2}{(x^2 - 1)^2} = \dfrac{25}{9} \\[1em] \Rightarrow \dfrac{x^2 + 1}{x^2 - 1} = \pm \dfrac{5}{3}

Considering, x2+1x2−1=53\dfrac{x^2 + 1}{x^2 - 1} = \dfrac{5}{3}

Applying componendo and dividendo, we get

⇒x2+1+x2−1x2+1−(x2−1)=5+35−3⇒2x22=82⇒x2=4⇒x=±2.\Rightarrow \dfrac{x^2 + 1 + x^2 - 1}{x^2 + 1 - (x^2 - 1)} = \dfrac{5 + 3}{5 - 3} \\[1em] \Rightarrow \dfrac{2x^2}{2} = \dfrac{8}{2} \\[1em] \Rightarrow x^2 = 4 \\[1em] \Rightarrow x = \pm 2.

Considering, x2+1x2−1=−53\dfrac{x^2 + 1}{x^2 - 1} = -\dfrac{5}{3}

Applying componendo and dividendo, we get

⇒x2+1+x2−1x2+1−(x2−1)=−5+3−5−3⇒2x22=−2−8⇒x2=14⇒x=14⇒x=±12.\Rightarrow \dfrac{x^2 + 1 + x^2 - 1}{x^2 + 1 - (x^2 - 1)} = \dfrac{-5 + 3}{-5 - 3} \\[1em] \Rightarrow \dfrac{2x^2}{2} = \dfrac{-2}{-8} \\[1em] \Rightarrow x^2 = \dfrac{1}{4} \\[1em] \Rightarrow x = \sqrt{\dfrac{1}{4}} \\[1em] \Rightarrow x = \pm \dfrac{1}{2}.

Hence, x = ±2 and ±12\pm \dfrac{1}{2}.

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