Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
4y² + 12y + 5

Answer 1

$$\begin{gathered} 4y^2+12y+5 \\ =4(y^2+3y+\frac{5}{4})[Makingthecoefficientof{y}^2=1] \\ =4[y^2+3y+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2+\frac{5}{4}][Addingandsubtracting{\left(\frac{3}{2}\right)}^2] \\ =4\left[\right(y+\frac{3}{2}{)}^2-\frac{9}{4}+\frac{5}{4}] \\ =4[(y+\frac{3}{2}{)}^2-1^2][Completingthesquare] \\ =4\left[\right(y+\frac{3}{2})-1]\left[\right(y+\frac{3}{2})+1] \\ =4(y+\frac{3}{2}-1)(y+\frac{3}{2}+1) \\ =4(y+\frac{1}{2})(y+\frac{5}{2}) \\ =(2y+1)(2y+5) \end{gathered}$$