Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
y² − 7y + 12

Answer 1

\[y^2 - 7y + 12\]
\[ = y^2 - 7y + \left( \frac{7}{2} \right)^2 - \left( \frac{7}{2} \right)^2 + 12 [\text{ Adding and subtracting }\left( \frac{7}{2} \right)^2 ]\]
\[ = (y - \frac{7}{2} )^2 - \frac{49}{4} + \frac{48}{4} [\text{ Completing the square }]\]
\[ = (y - \frac{7}{2} )^2 - \frac{1}{4} \]
\[ = (y - \frac{7}{2} )^2 - \left( \frac{1}{2} \right)^2 \]
\[ = [(y - \frac{7}{2}) - \frac{1}{2}][(y - \frac{7}{2}) + \frac{1}{2}]\]
\[ = (y - \frac{7}{2} - \frac{1}{2})(y - \frac{7}{2} + \frac{1}{2})\]
\[ = (y - 4)(y - 3)\]