The LessonCompleting 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 EquationsLet'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 ItThe 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 GeometryThe 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|
|x2 + 4x + 22 − 22 = (x + 2)2 − 22||Subtract 22 from both sides|
|x2 + 4x = (x + 2)2 − 22|