Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² − 14a − 51

Answer 1

$$\begin{gathered} a^2-14a-51 \\ =a^2-14a+{\left(\frac{14}{2}\right)}^2-{\left(\frac{14}{2}\right)}^2-51[Addingandsubtracting{\left(\frac{14}{2}\right)}^2,thatis,7^2] \\ =a^2-14a+7^2-7^2-51 \\ =(a-7{)}^2-100[Completingthesquare] \\ =(a-7{)}^2-{10}^2 \\ =\left[\right(a-7)-10]\left[\right(a-7)+10] \\ =(a-7-10)(a-7+10) \\ =(a-17)(a+3) \end{gathered}$$