The Lesson
Completing the square is a way of simplifying a
quadratic equation.
From
perfect square trinomials, we have seen that a
squared binomial expands to a quadratic equation:
If we have the first two
terms in a quadratic equation (the
x2 and
x terms), we can write it as a squared binomial minus a number:
Understanding Completing the Square Using Equations
Let's look at the patterns in the equations:
Halve the Number in Front of the x...
The number in the brackets (
2) is
half the number in front of the
x (
4).
...Then Square and Subtract It
The number being
subtracted from the squared brackets (
2) is
half the number in front of the
x (
4) squared (
22).
Understanding Completing the Square Using Geometry
The
area of the square plus the
area of the rectangle below equals
x2 + 4x:
Split the rectangle in half. Instead of one rectangle with an area of
4x, there are two rectangles each with an area of
2x:
Place the two rectangles by the side of the square:
We almost have a square. If we place a small square of area
22 in the space... we will
complete the square:
Each side of this larger square is
x + 2. Its area is
(x + 2)2.
By adding up the areas we see that:
| x2 + 2x + 2x + 22 = (x + 2)2 |
4 small areas = 1 big area |
| x2 + 2x + 2x + 22 = (x + 2)2 |
Add like terms |
| x2 + 4x + 22 = (x + 2)2 |
|
The original area was
x2 + 4x not
x2 + 4x + 22.
We
added 22 to complete the square. To balance the equation, we must subtract
22 from both sides:
| x2 + 4x + 22 − 22 = (x + 2)2 − 22 |
Subtract 22 from both sides |
| x2 + 4x = (x + 2)2 − 22 |
|
Completing the Square in General
In general, we complete the square as:
Lesson Slides
Completing the square works when the terms in the squared binomial are subtracted from each other.
The slider below shows another real example of completing the square.
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