Question : If $2x^2+5 x+1=0$, then one of the values of $x-\frac{1}{2 x}$ is:
Option 1: $\frac{\sqrt{17}}{2}$
Option 2: $\frac{13}{2}$
Option 3: $\frac{5}{2}$
Option 4: $\frac{\sqrt{13}}{2}$
Correct Answer: $\frac{\sqrt{17}}{2}$
Solution :
$2 x^2+5 x+1=0$
Dividing the number by $2x$, we get,
$⇒x+\frac{5x}{2x}+\frac{1}{2x}$ = 0
$\therefore x+\frac{1}{2x} = -\frac{5}{2}$
Using the formula $(a-b)^2 = (a+b)^2-4ab$, we get
$(x-\frac{1}{2x})^2 = (x+\frac{1}{2x})^2 - 4.x.\frac{1}{2x}$
$⇒(x-\frac{1}{2x})^2 = (-\frac{5}{2})^2-2$
$⇒(x-\frac{1}{2x})^2 = \frac{25}{4}-2$
$\therefore(x-\frac{1}{2x}) =\frac{\sqrt{17}}{2}$
Hence, the correct answer is $\frac{\sqrt{17}}{2}$.
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