Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
4x² − 12x + 5

Answer 1

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