Lesson What is Completing The Square? With Visual Explanation

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About josgarithmetic: Academic and job experience with beginning & intermediate Algebra. Tutorial help mostly for Basic Math and up through intermediate algebra.

COMPLETING THE SQUARE

When some problems or exercises are analyzed, sometimes quadratic expressions occur, such as anything of the form, x%5E2%2Bbx%2Bc

. The quadratic expression might be factorable as a perfect square, or it might not be. We cannot always rely on the expression to be a perfect square.

Examples of pefect square quadratics are ----
x%5E2%2B10x%2B25=%28x%2B5%29%5E2

;

x%5E2%2B%283%261%2F2%29x%2B3%261%2F16=%28x%2B1%263%2F4%29%5E2

;

An example of a quadratic which is not a perfect square----
x%5E2-12x%2B33

.
This is not factorable into any %28x%2Bk%29%5E2

WHAT IS COMPLETING THE SQUARE?

A number, c, may be added to form a quadratic expression x%5E2%2Bbx%2Bc

, and this quadratic may be unfactorable into any perfect square. We can still do something with the x%5E2%2Bbx

part.

A rectangle may have a length in one direction, x, and a possibly longer length in direction at right angle to it, x+b. The area of this rectangle is x%28x%2Bb%29

This rectangle is composed of a square of area x%5E2

, and a rectangular piece, of area x%2Ab

.
The area of the original entire rectangle is the sum of these two parts, x%5E2%2Bbx

;

Here shows the rectangular part of the whole figure cut into two equal pieces, each of length b%2F2

One of those halves can be moved to a different place on the figure as shown.

The total area of this modified figure is still the same as it was before. This is still an area of x%5E2%2Bbx=x%28x%2Bb%29

. It just appears different now.

A square corner piece would make this figure into a square shape, and if this piece were included here, forming a complete square, we would also be able to calculate the perfect square factorization. That is not exactly in fact what is wanted.

The areas of each individual piece of this figure, INCLUDING the missing square piece at the corner, can be formed. What are they?

The area pieces for our rearranged figure are x%5E2

, and another x%28b%2F2%29

.
The area for the missing part which would COMPLETE THE SQUARE is %28b%2F2%29%28b%2F2%29=%28b%2F2%29%5E2

.

Checking to see how the original pieces of area work, this area is x%5E2%2B%28b%2F2%29x%2B%28b%2F2%29x

Which is the same as what we started with for our original rectangle.

Now what can we do with the term for the missing square piece, %28b%2F2%29%5E2

?

Looking again at the non-factorable example of x%5E2-12x%2B33

, the missing square term is %28-12%2F2%29%5E2=36

. We can keep the same area if we BOTH ADD AND SUBTRACT 36 to this quadratic expression. We will not have a pefect square result, but still we will have a way to manage the quadratic expression.
x%5E2-12x%2B33%2B36-36

Meaning these represent equal areas, or equal numbers:
highlight_green%28x%5E2-12x%2B33=%28x-6%29%5E2-3%29

.

HERE IS COMPLETING THE SQUARE FOR A GENERAL QUADRATIC EXPRESSION
x%5E2%2Bbx%2Bc

, a general quadratic expression, assumed as not a perfect square.
Missing square term for x%5E2%2Bbx

. Add and Subtract this term.
x%5E2%2Bbx%2Bc%2B%28b%2F2%29%5E2-%28b%2F2%29%5E2

x%5E2%2Bbx%2B%28b%2F2%29%5E2%2Bc-%28b%2F2%29%5E2

highlight_green%28%28x%2Bb%2F2%29%5E2%2Bc-b%5E2%2F4%29

which may be further worked to
highlight_green%28%28x%2Bb%2F2%29%5E2%2B%284c-b%5E2%29%2F4%29

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