Completing the Square

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

Worked Example 7.3.1:

Solve the following:

i. x² + 5x
ii. y² + 9y
iii.  h² – 3h 
iv. x² + 7x 
v. x² – 10x 
vi. \( \scriptsize y^2 \: – \: \normalsize \frac{2y}{3} \)

Solution

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. \( \scriptsize y^2 \: – \: \normalsize \frac{2y}{3} \)

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

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

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

⇒ \( \frac{1}{9} \) must be added to make it a perfect square