Question

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

Answer 1

$$\begin{gathered} y^2-7y+12 \\ =y^2-7y+{\left(\frac{7}{2}\right)}^2-{\left(\frac{7}{2}\right)}^2+12[Addingandsubtracting{\left(\frac{7}{2}\right)}^2] \\ =(y-\frac{7}{2}{)}^2-\frac{49}{4}+\frac{48}{4}[Completingthesquare] \\ =(y-\frac{7}{2}{)}^2-\frac{1}{4} \\ =(y-\frac{7}{2}{)}^2-{\left(\frac{1}{2}\right)}^2 \\ =[(y-\frac{7}{2})-\frac{1}{2}\left]\right[(y-\frac{7}{2})+\frac{1}{2}] \\ =(y-\frac{7}{2}-\frac{1}{2}\left)\right(y-\frac{7}{2}+\frac{1}{2}) \\ =(y-4\left)\right(y-3) \end{gathered}$$