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Mathematics

The sum of a number and its reciprocal is 4.25. Find the number.

Quadratic Equations

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Answer

Let the number be x.

According to question,

⇒x+1x=4.25x+1x=425100x2+1x=1744(x2+1)=17x4x2+4=17x4x2−17x+4=04x2−16x−x+4=04x(x−4)−1(x−4)=0(4x−1)(x−4)=0(4x−1)=0 or (x−4)=0x=14 or x=4.\Rightarrow x + \dfrac{1}{x} = 4.25 \\[1em] x + \dfrac{1}{x} = \dfrac{425}{100} \\[1em] \dfrac{x^2 + 1}{x} = \dfrac{17}{4} \\[1em] 4(x^2 + 1) = 17x \\[1em] 4x^2 + 4 = 17x \\[1em] 4x^2 - 17x + 4 = 0 \\[1em] 4x^2 - 16x - x + 4 = 0 \\[1em] 4x(x - 4) - 1(x - 4) = 0 \\[1em] (4x - 1)(x - 4) = 0 \\[1em] (4x - 1) = 0 \text{ or } (x - 4) = 0 \\[1em] x = \dfrac{1}{4} \text{ or } x = 4.

Hence, number = 4 or 14\dfrac{1}{4}.

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