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Mathematics

Solve the following equation by factorisation:

x + 1x\dfrac{1}{x} = 2.5

Quadratic Equations

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Answer

x+1x=2.5⇒x2+1x=2510⇒10(x2+1)=25x⇒10x2+10=25x⇒10x2−25x+10=0⇒10x2−20x−5x+10=0⇒10x(x−2)−5(x−2)=0⇒(10x−5)(x−2)=0⇒(10x−5)=0 or x−2=0⇒10x=5 or x=2⇒x=12 or x=2.x + \dfrac{1}{x} = 2.5 \\[1em] \Rightarrow \dfrac{x^2 + 1}{x} = \dfrac{25}{10} \\[1em] \Rightarrow 10(x^2 + 1) = 25x \\[1em] \Rightarrow 10x^2 + 10 = 25x \\[1em] \Rightarrow 10x^2 - 25x + 10 = 0 \\[1em] \Rightarrow 10x^2 - 20x - 5x + 10 = 0 \\[1em] \Rightarrow 10x(x - 2) - 5(x - 2) = 0 \\[1em] \Rightarrow (10x - 5)(x - 2) = 0 \\[1em] \Rightarrow (10x - 5) = 0 \text{ or } x - 2 = 0 \\[1em] \Rightarrow 10x = 5 \text{ or } x = 2 \\[1em] \Rightarrow x = \dfrac{1}{2} \text{ or } x = 2.

Hence, x = 12\dfrac{1}{2} or x = 2.

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