Question

9x^2+3x+5=0 Find the answer by completing the square method Roots are wanted

Answer 1

Answer

No real roots

Given

9x² + 3x + 5 = 0

To Find

Solve equation by Completing the square method

Solution

$\rm 9x^2+3x+5=0\\\\\implies \rm \dfrac{9x^2}{9}+\dfrac{3x}{9}+\dfrac{5}{9}=\dfrac{0}{9}\\\\\implies \rm x^2+\dfrac{x}{3}+\dfrac{5}{9}=0\\\\\implies \rm x^2+2.x.\dfrac{1}{6}=-\dfrac{5}{9}\\\\\implies \rm x^2+2.x.\dfrac{1}{6}+\bigg(\dfrac{1}{6}\bigg)^2-\bigg(\dfrac{1}{6}\bigg)^2=-\dfrac{5}{9}\\\\\implies \rm \bigg(x+\dfrac{1}{6}\bigg)^2=\dfrac{1}{36}-\dfrac{5}{9}\\\\\implies \rm \bigg(x+\dfrac{1}{6}\bigg)^2=-\dfrac{19}{36}$

Since square of any number can't be negative .

So , roots doesn't exist . They are complex .

We can find roots . But they are in complex form .

$\implies \rm x+\dfrac{1}{6}=\dfrac{\pm \sqrt{19}\ i}{6}\\\\\implies \rm x=\dfrac{\pm\sqrt{19}\ i-1}{6}$