Lesson 7, Topic 3
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# Completing the Square

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To make a quadratic expression a perfect square, add half the coefficient of x (or whatever letters are used) and square the result.

i. x2 + 5x

Step 1: half the coefficient of  x

$$\frac{5}{2}$$

Step 2:  square the result

$$\scriptsize x^2 + \normalsize {\left(\frac{5}{2}\right)}^2 \\ = \left ( \scriptsize x + \normalsize \frac{5}{2} \right)^2$$ $${\left(\frac{5}{2}\right)}^2 = \frac{25}{4}$$

$$\frac{25}{4}$$ must be added to make it a perfect square

ii. y2 + 9y

= $$\scriptsize y^2 + \normalsize {\left(\frac{9}{2}\right)}^2 \\ = \left ( \scriptsize y + \normalsize \frac{9}{2} \right)^2$$

$$\frac{81}{4}$$ must be added to make it a perfect square

iii.  h2 – 3h

= $$\scriptsize h^2 \: – \: \normalsize {\left(\frac{3}{2}\right)}^2 \\ = \left ( \scriptsize h \: – \: \normalsize \frac{3}{2} \right)^2$$

$$\frac{9}{4}$$ must be added to make it a perfect square

iv. x2 + 7x

= $$\scriptsize x^2 + \normalsize {\left(\frac{7}{2}\right)}^2 \\ = \left ( \scriptsize x + \normalsize \frac{7}{2} \right)^2$$

$$\frac{49}{4}$$ must be added to make it a perfect square

v. x2 – 10x

= $$\scriptsize x^2 + \normalsize {\left(\frac{10}{2}\right)}^2 \\ = \left ( \scriptsize x + \normalsize \frac{10}{2} \right)^2$$

$$\frac{100}{4} \: \scriptsize = 25$$ must be added to make it a perfect square

vi. $$\scriptsize y^2 \: – \: \normalsize \frac{2y}{3}$$

= $$\scriptsize y^2 \: – \: \left (\normalsize \frac{\frac{2y}{3}}{2} \right)$$

= $$\scriptsize y^2 \: – \: \left (\normalsize \frac{2}{3} \: \times \: \frac{1}{2} \right)$$

= $$\scriptsize y^2 \: – \: \normalsize {\left(\frac{2}{6}\right)}^2 \\ = \scriptsize y^2 \: – \: \normalsize {\left(\frac{1}{3}\right)}^2 \\ \left (\scriptsize y \: – \: \normalsize \frac{1}{3}\right)^2$$

$$\frac{1}{9}$$ must be added to make it a perfect square error: