Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
q2 − 10q + 21

Answer 1

$$\begin{gathered} {\mathrm{q}}^2-10\mathrm{q}+21 \\ ={\mathrm{q}}^2-10\mathrm{q}+{\left(\frac{10}{2}\right)}^2-{\left(\frac{10}{2}\right)}^2+21[\mathrm{Adding}\mathrm{and}\mathrm{subtracting}{\left(\frac{10}{2}\right)}^2,\mathrm{that}\mathrm{is},5^2] \\ ={\mathrm{q}}^2-2\times \mathrm{q}\times 5+5^2-5^2+21 \\ =(\mathrm{q}-5{)}^2-4[\mathrm{Completing}\mathrm{the}\mathrm{square}] \\ =(\mathrm{q}-5{)}^2-2^2 \\ =\left[\right(\mathrm{q}-5)-2]\left[\right(\mathrm{q}-5)+2] \\ =(\mathrm{q}-5-2)(\mathrm{q}-5+2) \\ =(\mathrm{q}-7)(\mathrm{q}-3) \end{gathered}$$