Solve the following equation:

$x + \dfrac{1}{x} = 25\dfrac{1}{{25}}$

Hint: First take LCM on the RHS part of the given equation and convert it to the quadratic equation. Later by factorizing find the values of x.

Complete step by step answer:

According to the question given equation is

$ \Rightarrow x + \dfrac{1}{x} = 25\dfrac{1}{{25}} $...................................(1)

Now, taking LCM on LHS and solving mixed fraction of RHS we get,

$ \Rightarrow \dfrac{{{x^2} + 1}}{x} = \dfrac{{626}}{{25}}$

Now, cross multiply LHS and RHS we get,

$\Rightarrow {\text{25}}\left( {{x^2} + 1} \right) = 626x $

Taking RHS to LHS we get a quadratic equation that is,

$\Rightarrow 2{\text{5}}{x^2} - 626x + 25 = 0$...........................................(2)

Splitting the middle term of equation 2 we get,

$ \Rightarrow 2{\text{5}}{x^2} - 625x - x + 25 = 0$

On, solving above equation we get,

$ \Rightarrow 2{\text{5}}x(x - 25) - 1(x - 25) = 0 $

$ \Rightarrow (2{\text{5}}x - 1)(x - 25) = 0$

So, value of x will be 25 or $\dfrac{1}{{25}} $

Hence, x = $25,\dfrac{1}{{25}}$

NOTE: - Whenever we come up with this type of problem then the best way is to cross - multiply both sides and get a quadratic equation and then solve that quadratic equation by splitting terms.