Which constant must be added and subtracted

Question:

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

(a) $\frac{1}{8}$

(b) $\frac{1}{64}$

(C) $\frac{1}{4}$

(d) $\frac{9}{64}$

Solution:

(b) Given equation is $9 x^{2}+\frac{3}{4} x-\sqrt{2}=0$.

$(3 x)^{2}+\frac{1}{4}(3 x)-\sqrt{2}=0$

On putting $3 x=y$, we have $y^{2}+\frac{1}{4} y-\sqrt{2}=0$

$y^{2}+\frac{1}{4} y+\left(\frac{1}{8}\right)^{2}-\left(\frac{1}{8}\right)^{2}-\sqrt{2}=0$

$\left(y+\frac{1}{8}\right)^{2}=\frac{1}{64}+\sqrt{2}$

$\left(y+\frac{1}{8}\right)^{2}=\frac{1+64 \cdot \sqrt{2}}{64}$

Thus, $\frac{1}{64}$ must be added and subtracted to solve the giveri equation.

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