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1.08 Quadratic
Equations
College
Algebra: One Step at a Time. Pages
122-124: #7
Pages
125 -129: #11, 12
NEW
PROBLEM: #7
Dr. Robert J. Rapalje
Seminole Community College
Sanford, FL 32773
When solving a quadratic
equation, there are several methods to consider. Some problems can be
solved best by one method, and other problems are best solved by different
methods.
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Methods of Solving Quadratic Equations |
- Factoring
- Quadratic Formula
- Completing the
Square
- Calculator Method
by Graphing
- Calculator
POLYSMLT
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Of course, the easiest way
to solve any quadratic equation is the method of factoring--that is, of
course, IF the equation can be factored. Not all quadratic equations can be
factored.
The most reliable method
is the quadratic formula:
If
 ,
then
 .
The quadratic formula ALWAYS
works!
However, while the quadratic
formula can be used to solve any problem, the completing the square method
is sometimes easier. Specifically, in order to solve by the completing the
square method, the coefficient of
 must
be 1 in order for “half and square” to work. Secondly, the completing the
square method works very nicely when the x coefficient is an even number (so
taking “half and square” doesn’t end up with horrible fractions to deal
with!).
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Summary for Completing the
Square
for
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- Recommended as the
best method whenever a=1 and
b = even number!
- Express the
equation with variable terms on the left and the number term on
the right side of the equation, in the form
 .
- Take half of the x
coefficient, and square, in order to decide what number to add
to each side of the equation to complete the square. That is,
“half, and square”.
- Express as a
binomial squared equal to a number.
- Take the square
root of each side of the equation—don’t forget the
 .
- Solve for x.
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Completing the Square
p. 124.
#7. Solve for x by
completing the square:
 .
Solution:
Set up for completing the
square by adding 15 to each side:
Next, take
“half and square”:
Square root both sides:
Simplify
 :
Add
+5 to each
side: 
NOTE: If you do NOT have POLYSMLT on your TI
83+ or TI 84 calculator in the APPS menu of your calculator, it can be
easily installed for you if you know someone who has it on their calculator!
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Check with POLYSMLT (TI
83+ or TI 84): |
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[APPS]
[POLYSMLT]
[ENTER] [ENTER] [ENTER]
Degree = [2] (This is the highest
power of x in the equation!)
Now, enter the coefficients, followed
by [ENTER]
[1] [ENTER]
[-10] [ENTER]
[-15] [ENTER]
[F5, SOLVE]
Answer: x = -1.32455532; x
= 11.32455532
Now calculate, and compare to the
calculator values from POLYSMLT:
 =
11.32455532
 =
-1.32455532
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If you have a TI 86 or TI 85, the program you need
is built into the calculator at [2nd] [POLY].
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Check with TI 86 or TI
85 |
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[2nd] [POLY]
Order = [2] (This is the highest
power of x in the equation!)
Now, enter the coefficients, followed
by [ENTER]
[1] [ENTER]
[-10] [ENTER]
[-15] [ENTER]
[F5, SOLVE]
Answer: x = -1.32455532; x
= 11.32455532
Now calculate, and compare to the
calculator values from [2nd] [POLY]:
=
11.32455532
=
-1.32455532
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Quadratic Formula
p. 129. # 11.
This is a quadratic equation, so the first step is to set it equal to zero,
preferably with a positive  term.
This can be done by moving everything to the right side, by adding and  to
each side.
This quadratic equation cannot be solved by factoring, so in this case, the
quadratic formula is the best choice of methods.
Remember that the solution to the quadratic equation: 
is given by the formula:
In the equation: 
Factor out
the common factor of 2 in the numerator, and reduce the fraction.
p. 129. # 12. 
The first
step is to multiply out the parentheses, and notice that it is a quadratic
equation. When you see this, you might as well go ahead and set it equal to
zero by adding +5 to each side of the equation.
This quadratic equation cannot be solved by factoring, so the quadratic
formula is the best method.
Remember that the solution to the quadratic equation:
is given by the formula:
In the equation: 
Factor out the common factor
of
2 in the numerator, and
reduce the fraction.
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