Links for Class:
http://math404.weebly.com/
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Completing The Square
Things to Do:
(b)
x + 12x + 36 = 17
2
2
1. WarmUp: Solve quadratic equations by using the
square root method
(a)
x + 8x + 16 = 25
2. Definition of completing the square
3. Six Steps for the completing the square algorithm
4. Examples of solving quadratic eqauations by using the
method of completing the square
(c)
x + 8x + 14 = 0
(d)
x + 12x + 7 = 0
(e)
x + 16x + 12 = 0
2
2
2
WarmUp:
2
(a)
x + 8x + 16 = 25
2
2
(x+4) = 25
(x+4) = 25
x + 4 = 5
Therefore, x = 1 or x = 9
2
(b)
x + 12x + 36 = 17
2
2
(x+6) = 17
(x+6) = 17
x + 6 = 17
Therefore, x = 6 + 17 or x = 6  17
Definition of completing the square:
Completing The Square is defined as a method that creates perfect square trinomials to solve quadratic equations when factoring is not possible.
Steps for the completing the square algorithm
Step 1:
Rewrite the equation so that the left side of the equation is in the form x + bx.
2
Examples:
(c)
x + 8x + 14 = 0
2
2
2
2
2
2
(d)
x + 12x + 7 = 0
2
2
2
2
2
2
Step 6:
(x+6) = 29
x + 6 = 29
Therefore, x = 6 + 29 or x = 6  29
(e)
x + 16x + 12 = 0
2
2
2
2
2
2
Step 6:
(x8) = 76
x  8 = 2 19
Therefore, x = 8 + 2 19 or x = 8  2 19
Recall:
Examples of representations of quadratic equations
Step 1
: Arrange the Algebra Tiles in the square
and add zero pairs to complete the square.
Directions for
Algebra Tiles Applet:
Step 2
: Add Algebra Tiles on both of the sides to
represent factors.
Homework:
In textbook, Section 6.4, Problems: 23, 24
Directions:
Solve each quadratic equation by electronically creating with virtual Algebra Tiles from The National Library of Virtual Manipulatives and then handdraw a geometrical representation for completing the square with virtual Algebra Tiles and add the needed zero pairs.
In textbook, Section 6.4, Problems: 27, 29, 31
Directions:
Solve each quadratic equation by using the traditional method of completing the square. If you want to receive full credit, be sure to show all of your work and do not forget any steps.
Due December 3, 2013
Step 1:
x + 8x = 14
Step 2:
( )(8) = 4
Step 3:
4 = 16
Step 4:
x + 8x + 16 = 14 + 16
Step 6:
(x+4) = 2
x + 4 = 2
Therefore, x = 4 + 2 or x = 4  2
Step 5:
(x+4) = 2
Step 1:
x + 12x = 7
Step 2:
( )(12) = 6
Step 3:
6 = 36
Step 4:
x + 12x + 36 = 7 + 36
Step 5:
(x+6) = 29
Step 1:
x  16x = 12
Step 2:
( )(16) = 8
Step 3:
(8) = 64
Step 4:
x  16x + 64 = 12 + 64
Step 5:
(x8) = 76
What if the quadratic expression is in the form x² + bx + c where c is an irrational root?
If the quadratic equation is in the form x² + bx + c, such that x² coefficient equals ±1 and b (the xcoefficient) is even, then the completing the square algorithm consists of six steps.
Step 2
: Solve for one half of b (the xcoefficient).
Step 3:
Square the result from step 2.
Step 4:
Add the result from step 3 to each side.
Step 5:
Factor the left side of the equation such that the left side is a perfect square.
Step 6:
Solve the equation by using the Square Root method.
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Completing The Square
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