(Image courtesy of a student collaborating with us; Jacob Reeves)

The third way to solve quadratic equations is to complete the square.

Example: X^2+8X+12=32

Step 1: Check to make sure that the first term only has an X^2 in it) Remember, measure twice, cut once!

Step 2: subtract number outside equal sign by the last number term (called a constant)

32-12=20 and 12-12=0 (or blank)

X^2+8X+___=12

Step 3: take the number in the middle (the one with the one x) divide it by 2 and multiply it by itself. That is called the MAGIC number.

(I included the magician hat as a way to remember what the number is.)

Example: 8/2=4. Then 4*4=16

then put that number in the empty place where the last term used to be. While leaving the other numbers alone.

X^2+8X+_16___=20

Step 4: subtract number outside the equal sign by the MAGIC number.

Ex: 20-16=4

now you have this:

2x^2+8x+16=4

step 5: take square root of first term and last term and keep the middle sign.

√2x^2=2x

√16=4

So it is: 2x+4

Step 6: You also take the square root of the number outside the equal sign.

√4=+2 and -2

Step 7: solve the two equations!

2x+4=2. And 2x+4=-2

step 8: write your answer (then plug in original formula to check if it comes out as zero)

x=-1 and x=-3

(Sometimes you may get a square root of a negative number. Normally that cannot happen but check the following meme below)

That is why in another section we will learn about the exception called “Imaginary Numbers”.

Fun Fact: The first to invent completing the square were the Babylonians though others later on realized how to do it. And that is why we have the Quadratic Formula