Tutoring high school math, this is a celebrated topic.  It even gets revisited at various levels of calculus. The math tutor introduces it, having just come in from shoveling the drive….

One of many situations in which you might need to complete the square is the following:

x^2 – 6x =17

It’s a quadratic; naturally, you want to subtract 17 from both sides and hopefully factor.

x^2 – 6x -17 = 0

However, you already know that x² – 6x -17 doesn’t factor. What might you do instead?

Completing the square is one option. To do so, leave 17 on the right side:

x^2 -6x =17

Step 1:

Focus on the coefficient of the x term: in this case, the -6. Specifically, take half of it, then square that.

-6÷2 = -3→(-3)^2=9

Step 2:

Add the result from Step 1 to both sides of the equation:

x^2 – 6x + 9 = 17 + 9

Step 3:

Notice that now, the left side does indeed factor. In this case, it factors to (x-3)(x-3), aka (x-3)². We rewrite the equation correspondingly:

(x-3)^2=26

Step 4:

With the left side in square form, we can square root both sides, to get

x-3=±√(26)

We add three to both sides, yielding

x=3+√(26) or x=3-√(26)

This is a light introduction to a topic that can be more complicated. However, it shows the usefulness and elegance of completing the square. More uses and complications of it will be visited in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.