Question

In order to solve the quadratic equation $4x^2-\sqrt{3}x-5=0$ by method of completing the square, without making coefficient of x² unity, we must add and subtract the number _________.

Answer 1

Let $4x^2-\sqrt{3}x-5=0$ be a quadratic equation.

$$\begin{gathered} 4x^2-\sqrt{3}x-5=0 \\ \Rightarrow {\left(2x\right)}^2-4\left(\frac{\sqrt{3}}{4}\right)x-5=0 \\ \Rightarrow {\left(2x\right)}^2-2\left(2x\right)\left(\frac{\sqrt{3}}{4}\right)-5=0 \\ \mathrm{To}\mathrm{make}\mathrm{it}\mathrm{a}\mathrm{complete}\mathrm{square}\mathrm{we}\mathrm{must}\mathrm{add}\mathrm{and}\mathrm{subtract}{\left(\frac{\sqrt{3}}{4}\right)}^2 \\ \Rightarrow {\left(2x\right)}^2-2\left(2x\right)\left(\frac{\sqrt{3}}{4}\right)+{\left(\frac{\sqrt{3}}{4}\right)}^2-{\left(\frac{\sqrt{3}}{4}\right)}^2-5=0 \\ \Rightarrow \left[{\left(2x\right)}^2-2\left(2x\right)\left(\frac{\sqrt{3}}{4}\right)+{\left(\frac{\sqrt{3}}{4}\right)}^2\right]-{\left(\frac{\sqrt{3}}{4}\right)}^2-5=0 \\ \Rightarrow {\left(2x-\frac{\sqrt{3}}{4}\right)}^2-{\left(\frac{\sqrt{3}}{4}\right)}^2-5=0 \end{gathered}$$

Hence, in order to solve the quadratic equation $4x^2-\sqrt{3}x-5=0$ by method of completing the square, without making coefficient of x² unity, we must add and subtract the number $\underline{\frac{3}{16}}$.