Question

Find the roots of $4x^2+3x+5=0$ by the method of completing the square.

Answer 1

Given equation

$4x^2+3x+5=0$

Dividing by 4

$\frac{4x^2}{4}+\frac{3x}{4}+\frac{5}{4}=0$

$x^2+\frac{3x}{4}+\frac{5}{4}=0$

We know that

$(a+b{)}^2=a^2+2ab+b^2$

Here $a=x$

$2ab=\frac{3x}{4}$

$2b=\frac{3}{4}$

$b=\frac{3}{8}$

Adding and subtracting ${\left(\frac{3}{8}\right)}^2$

$x^2+\frac{3x}{4}+\frac{5}{4}+{\left(\frac{3}{8}\right)}^2-{\left(\frac{3}{8}\right)}^2=0$

${(x+\frac{3}{8})}^2+\frac{5}{4}-{\left(\frac{3}{8}\right)}^2=0$

${(x+\frac{3}{8})}^2=\frac{9}{64}-\frac{5}{4}$

${(x+\frac{3}{8})}^2=\frac{9-80}{64}$

${(x+\frac{3}{8})}^2=\frac{-71}{64}$

Since square of any number cannot be negative

So, answer does not exist.