Question

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

$y^2-7y+12$

Answer 1

Given: $y^2-7y+12$

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

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

i.e., ${\left(\frac{-7}{2}\right)}^2={\left(\frac{7}{2}\right)}^2$

$\Rightarrow y^2–7y+12$

$=y^2–7y+{\left(\frac{7}{2}\right)}^2–{\left(\frac{7}{2}\right)}^2+12$

$=y^2–7y+{\left(\frac{7}{2}\right)}^2–\frac{49}{4}+12$

$=y^2–7y+{\left(\frac{7}{2}\right)}^2–\frac{1}{4}$

$={(y-\frac{7}{2})}^2–{\left(\frac{1}{2}\right)}^2$

$=(y-\frac{7}{2}-\frac{1}{2})(y-\frac{7}{2}+\frac{1}{2})$

$=(y-\frac{8}{2})(y-\frac{6}{2})$

$=(y–4)(y–3)$

Hence, $y^2-7y+12=(y–4)(y–3)$