Question

Which constant should be added and subtracted to solve the quadratic equation $4x^2-\sqrt{3}x-5=0$ by the method of completing the square?
(a) $\frac{9}{10}$
(b) $\frac{3}{16}$
(c) $\frac{3}{4}$
(d) $\frac{\sqrt{3}}{4}$

Answer 1

(b) $\frac{3}{16}$

$$\begin{gathered} \mathrm{Given}: \\ 4x^2-\sqrt{3}x-5=0 \\ \Rightarrow {\left(\text{2}x\right)}^2-2\times 2x\times \frac{\sqrt{3}}{4}+{\left(\frac{\sqrt{3}}{4}\right)}^2-{\left(\frac{\sqrt{3}}{4}\right)}^2-5=0 \\ \Rightarrow {\left(2x-\frac{\sqrt{3}}{4}\right)}^2+\frac{3}{16}-5=0 \\ \text{Clearly, if}\frac{3}{16}\text{is added to the equation, the equation becomes a perfect square}\text{.} \end{gathered}$$