Question

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

Answer 1

\[4 y^2 + 12y + 5\]
\[ = 4( y^2 + 3y + \frac{5}{4}) [\text{ Making the coefficient of }y^2 = 1]\]
\[ = 4[ y^2 + 3y + \left( \frac{3}{2} \right)^2 - \left( \frac{3}{2} \right)^2 + \frac{5}{4}] [\text{ Adding and subtracting }\left( \frac{3}{2} \right)^2 ]\]
\[ = 4[(y + \frac{3}{2} )^2 - \frac{9}{4} + \frac{5}{4}]\]
\[ = 4[(y + \frac{3}{2} )^2 - 1^2 ] [\text{ Completing the square }]\]
\[ = 4[(y + \frac{3}{2}) - 1][(y + \frac{3}{2}) + 1]\]
\[ = 4(y + \frac{3}{2} - 1)(y + \frac{3}{2} + 1)\]
\[ = 4(y + \frac{1}{2})(y + \frac{5}{2})\]
\[ = (2y + 1)(2y + 5)\]