If 2x² - 7x + 5 = 0, then what is the value of $${x^2} + \frac{{25}}{{4{x^2}}}?$$

Solution (By Examveda Team)

$$\eqalign{ & 2{x^2} - 7x + 5 = 0 \cr & 2x + \frac{5}{x} = 7 \cr & x + \frac{5}{{2x}} = \frac{7}{2} \cr & {\left( {x + \frac{5}{{2x}}} \right)^2} = {\left( {\frac{7}{2}} \right)^2} \cr & {x^2} + \frac{{25}}{{4{x^2}}} + 2 \times x \times \frac{5}{{2x}} = \frac{{49}}{4} \cr & {x^2} + \frac{{25}}{{4{x^2}}} = \frac{{49}}{4} - 5 \cr & {x^2} + \frac{{25}}{{4{x^2}}} = \frac{{29}}{4} \cr & {x^2} + \frac{{25}}{{4{x^2}}} = 7\frac{1}{4} \cr} $$