completing the square (geometrically)
DESCRIPTION
Most people teach algebra the same old boring way, using geometry in algebra is a great way to have students visualize what they are doing and most people have no idea why they complete a square.TRANSCRIPT
Completing the Square
Review… how do you solve b ? x + =5 22g 0
25
x
x
Left side of this equation is a perfect square… Geometrically this is as follows, x + 5 x 5 x x + 5 Break it up like this 5 Look at the areas x 5 x x 2 5x
5 5x 25 So, b ! x x x+ = + +5 102 2g Now, let’s take a look at geometrically. x 6 x2 6+ x + x Geometrically, to make a square, we need to cut the 6x piece in half and separate as follows: x2 6+ 3 3 3 3 x x and x Slide one of the 3x pieces against the side of the x 2 piece. Rotate the other and slide against the bottom of the x 2 piece like this… x2 3x 3x Notice the missing area? How much is it? __________ So, in order to complete the square you must add 9 to . x2 6+This means that , or geometrically we have, x x x2 26 9 3+ + = +b g x 3 x x 2 3x 3 3x 9
Algebraically, how do we accomplish completing the square? First notice, the coefficient of x2 needs to be one (1). If it is not, you must first divide through to make the coefficient equal to 1. Then… 1. Take 1 the coefficient of x 2
2. Square the value from step 1 and add this to the original expression. 3. The new expression now factors to a perfect square (x + d) where d is the value obtained in step 1. Ex. Complete the square:
x x
x x
x
2
2
2
2
2 1
1
12
2 1
1 12
+ +
+ +
+
=
=
a f
b g
x x
x x
x
2
2
2
3
94
32
1
23
3
2
3
2
9
4
2
+ +
( )
FH IK+ +
+FHGIKJ
=
=
You can use this concept to solve equations as well. Consider . Solve for x by completing the square. 2 6 32x x− + = 0First, make the coefficient of equal to 1, by dividing all the terms by 2 to obtain, x2
x x2 3
23 0− + = Next, get all the x terms on the left – side and the constant terms on the right.
x x2 323− = −
Thirdly, complete the square on the left.
x x2 323− + = − +
12
3 32
32
94
2
− = −
−FHGIKJ =
b g
This means in order to complete the square, you need to add 9
4 to my left side of the equation. You also
need to add 94 to the other side of the equation. Why? You should then have,
x x2 9
432
943− + = − +
Factoring the left side, and simplifying the right gives you,
x + =32
2 34b g
You should now solve like you did the review problem on the other side of the page in order to verify the solutions given below. The solutions are
x = ±3 32
≈ ≈2 37 0 63. .or