Question

To solve the quadratic equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$by the method of completing the square, without making making the coefficient of x² unity, we must add and subtract _________.

Answer 1

Let $9x^2+\frac{3}{4}x-\sqrt{2}=0$ be a quadratic equation.

$$\begin{gathered} 9x^2+\frac{3}{4}x-\sqrt{2}=0 \\ \Rightarrow {\left(3x\right)}^2+2\left(\frac{3}{8}\right)x-\sqrt{2}=0 \\ \Rightarrow {\left(3x\right)}^2+2\left(3x\right)\left(\frac{1}{8}\right)-\sqrt{2}=0 \\ \mathrm{To}\mathrm{make}\mathrm{it}\mathrm{a}\mathrm{complete}\mathrm{square}\mathrm{we}\mathrm{must}\mathrm{add}\mathrm{and}\mathrm{subtract}{\left(\frac{1}{8}\right)}^2 \\ \Rightarrow {\left(3x\right)}^2+2\left(3x\right)\left(\frac{1}{8}\right)+{\left(\frac{1}{8}\right)}^2-{\left(\frac{1}{8}\right)}^2-\sqrt{2}=0 \\ \Rightarrow \left[{\left(3x\right)}^2+2\left(3x\right)\left(\frac{1}{8}\right)+{\left(\frac{1}{8}\right)}^2\right]-{\left(\frac{1}{8}\right)}^2-\sqrt{2}=0 \\ \Rightarrow {\left(3x+\frac{1}{8}\right)}^2-{\left(\frac{1}{8}\right)}^2-\sqrt{2}=0 \end{gathered}$$

Hence, to solve the quadratic equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$by the method of completing the square, without making making the coefficient of x² unity, we must add and subtract $\underline{\frac{1}{64}}$.