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Question
Factorize each of the following quadratic polynomials by using the method of completing the square:
x2 + 12x + 20
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Solution
\[x^2 + 12x + 20\]
\[ = x^2 + 12x + \left( \frac{12}{2} \right)^2 - \left( \frac{12}{2} \right)^2 + 20 [\text{ Adding and subtracting }\left( \frac{12}{2} \right)^2 ,\text{ that is }, 6^2 ]\]
\[ = x^2 + 12x + 6^2 - 6^2 + 20\]
\[ = (x + 6 )^2 - 16 [\text{ Completing the square }]\]
\[ = (x + 6 )^2 - 4^2 \]
\[ = [(x + 6) - 4][(x + 6) + 4]\]
\[ = (x + 6 - 4)(x + 6 + 4)\]
\[ = (x + 2)(x + 10)\]
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