**A REVIEW OF SQUARE COMPLETION**

Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.

Square Completionis carried out to achieve perfect square trinomials of the following form:or

Please not that

equals

equals

IMPORTANT: Notice that the third term of either of the two trinomials above is found by dividing the coefficient

of the second term (2ax-term) byand raising the result to the second power2

The Square Completion Strategy Given a Quadratic Equation in Two Variables

- Isolate or on one side of the equation.
- If necessary, change the coefficient of to
.1- Find by dividing the coefficient of the
x-term, which is, by2aand raising the result to the second power.2- You then add to both sides of the equation! This ensures that equality is preserved. See WARNING below!

WARNING: Should you have changed the coefficient of by factoring out a number, sayk, you must add/subtract !Example 1:

Given the equation complete the square on the right side

Step 1:

Isolate the terms containing

on one side of the equation.x

y = x^{2}xStep 2:

Let's multiply both sides of the equation by

to change the coefficient of to2.1

2y + 1 = x^{2}+ 2xStep 3:

Find by dividing the coefficient of the

x-term, which is, by2and raising the result to the second power.2That is,

Step 4:

We then add

form Step 3 to both sides of the equation! This ensures that equality is preserved.1

2y + 1 + 1 = x^{2}+ 2x + 1

2y + 2 = x^{2}+ 2x + 1and

2y + 2 = (x + 1)^{2}As requested in the instructions, we completed the square on the right side of the equation.

Example 2:

Given the equation

complete the square on the right side.x = 2y^{2}12y + 7Please note that we will use the above strategy, however, we will separate the

y-terms from thexand the constant.Step 1:

Isolate the terms containing

on one side of the equation.y

x 7 = 2y^{2}12yStep 2:

Let's factor

out of the right side to change the coefficient of to2.1

)x 7 = 2(y^{2}6yStep 3:

Find by dividing the coefficient of the

y-term, which is, by6and raising the result to the second power.2That is, .

Step 4:

So do we now add

from Step 3 to both sides of the equation?9NO!Please note that we CANNOT just add a

to the left side. This would change the value of the expression! Actually we must add9because that's what we did on the right side too! Please study the following equation carefully!2(9)

x 7 + 2(9) = 2(y^{2}6y + 9)Are you noticing that we actually added

to the right side?2(9)and

x + 11 = 2(y 3)^{2}As requested in the instructions, we completed the square on the right side of the equation.