Question

Factorize the following quadratic polynomial by using the method of completing the square:

$4x^2-12x+5$

Answer 1

Given: $4x^2-12x+5$

$=4(x^2–3x+\frac{5}{4})$

Since, the coefficient of $x^2$ is unity.

$\therefore$ Adding and subtracting $(\frac{1}{2}\times$ coefficient of $x{)}^2$

i.e., ${(-\frac{3}{2})}^2={\left(\frac{3}{2}\right)}^2$

$\Rightarrow 4(x^2–3x+\frac{5}{4})$

$=4[x^2–3x+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2+\frac{5}{4}]$

$=4[x^2–3x+{\left(\frac{3}{2}\right)}^2-\frac{9}{4}+\frac{5}{4}]$

$=4[{(x–\frac{3}{2})}^2–1^2]$

$=4(x–\frac{3}{2}-1)(x–\frac{3}{2}+1)$

$=4(x–\frac{5}{2})(x–\frac{1}{2})$

$=4\left(\frac{2x–5}{2}\right)\left(\frac{2x-1}{2}\right)$

$=(2x–5)(2x–1)$

Hence, $4x^2-12x+5=(2x–5)(2x–1)$