Question

Which constant must be added and subtracted to solve the quadratic equation $9x^2+\frac{3}{4}x+2=0$by the method of completing the square?

Answer 1

Completing square method:

The given expression is: $9x^2+\frac{3}{4}x+2=0$

Dividing by 9 on both sides, $x^2+\frac{x}{12}+\frac{2}{9}=0$
$x^2+\frac{x}{12}=-\frac{2}{9}$
Third term =$(\frac{1}{2}\times coefficientofx{)}^2$

$$\begin{gathered} =(\frac{1}{12}{)}^2 \\ =(\frac{1}{24}{)}^2 \\ =\left(\frac{1}{576}\right) \end{gathered}$$

Adding $\frac{1}{576}$on both sides,

$$\begin{gathered} x^2+\frac{x}{12}+\frac{1}{576}=\frac{1}{576}-\frac{2}{9} \\ \Rightarrow (x+\frac{1}{24}{)}^2=\frac{1}{576}-\frac{2}{9} \end{gathered}$$

Hence,$\frac{1}{576}$is added and subtracted to make it perfect square.