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Question

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

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Solution

${\mathrm{q}}^{2}-10\mathrm{q}+21\phantom{\rule{0ex}{0ex}}={\mathrm{q}}^{2}-10\mathrm{q}+{\left(\frac{10}{2}\right)}^{2}-{\left(\frac{10}{2}\right)}^{2}+21\left[\mathrm{Adding}\mathrm{and}\mathrm{subtracting}{\left(\frac{10}{2}\right)}^{2},\mathrm{that}\mathrm{is},{5}^{2}\right]\phantom{\rule{0ex}{0ex}}={\mathrm{q}}^{2}-2×\mathrm{q}×5+{5}^{2}-{5}^{2}+21\phantom{\rule{0ex}{0ex}}=\left(\mathrm{q}-5{\right)}^{2}-4\left[\mathrm{Completing}\mathrm{the}\mathrm{square}\right]\phantom{\rule{0ex}{0ex}}=\left(\mathrm{q}-5{\right)}^{2}-{2}^{2}\phantom{\rule{0ex}{0ex}}=\left[\left(\mathrm{q}-5\right)-2\right]\left[\left(\mathrm{q}-5\right)+2\right]\phantom{\rule{0ex}{0ex}}=\left(\mathrm{q}-5-2\right)\left(\mathrm{q}-5+2\right)\phantom{\rule{0ex}{0ex}}=\left(\mathrm{q}-7\right)\left(\mathrm{q}-3\right)\phantom{\rule{0ex}{0ex}}$

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