5.07
Quadratic Equation
from Basic Algebra: One Step at
a Time © 2002-2011
P. 447-452
Dr. Robert J. Rapalje
Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE
END OF THIS PAGE
In Chapter 2,
certain quadratic equations (equations involving a variable raised to the
second power) were solved by factoring. In that chapter, one important
fact about quadratic equations was neglected. That is, not all quadratic
equations can be solved by factoring. Since all quadratic equations that
cannot be factored involve radicals, it was necessary to save this topic
until now. ALL quadratic equations can be solved using what is known as
the quadratic formula. This formula will be derived in a higher algebra
course by a method called completing the square. In this first level of
algebra, the formula is presented and used without proof.
The general form
of the quadratic equation is ax2+bx+c
= 0,
where a, b, and c represent any real numbers. Notice that a is
always the coefficient of x2,
b is always the coefficient of x, and c is always the constant
coefficient. The solution to this equation is given by the
World Famous Quadratic Formula!!
If this is the
first time you have ever seen this formula, it can be rather
intimidating. However, it is not nearly as bad as it looks. (It is
really as easy as a, b, c!) If you are wondering, Do I have to
memorize this formula?, the answer is, No! You will learn it very well
just by doing all the homework. If you are not going to do the homework,
then Yes! You need to memorize it!
The quadratic
formula may be used to solve any quadratic equation, even if the
equation can be factored. However, it is usually best to use factoring if
the equation factors and use the quadratic formula otherwise. Please
note that, while the use of the quadratic formula on the next pages
is not as easy as the factoring method, it does work on all
problems, and it is much easier than it may at first appear.
Just follow the outline that is given on the next page.
447
GUIDELINES FOR USING QUADRATIC FORMULA
1. The
equation should always be set equal to zero.
2.
Arrange the terms
in descending powers of the variable.
3.
Identify the a,
b,
and c:
The coefficient of x2
is a,
the coefficient of x is b,
and the constant term is c.
4.
It is preferred
(not required!) that a be positive.
5.
Write down the
formula.
6.
Substitute a,
b,
and c.
7.
Simplify the
radical, simplify and reduce the fraction if possible.
EXAMPLE
1. Solve for
x: x2
+ 5x - 2 = 0 . EXAMPLE 2.
Solve for x: x2
+ 5 = 6x .
Solution:
Equation is already set = 0 . Solution:
Set equal to zero: x2 - 6x + 5 = 0.
a = 1 b = 5 c =
-2 a = 1 b = -6 c = 5
Example 1: Fraction
cannot be reduced!
Notice that
in Example 1, the answer contains a radical, whereas in Example 2, you
obtained the square root of a perfect square. This perfect square in
Example 2 means that it could have been solved by factoring, whereas
Example 1 (not a perfect square) could not be solved by factoring.
448
EXERCISES.
Solve the following quadratic equations using the
quadratic formula.
1. x2
+ 3x - 2 = 0 2. x2
+ 5 = 5x
Equation is already set = 0. Set
equal to zero: x2 ____x + ____= 0
a=____ b=____ c=____ a=____
b=____ c=____
Write formula: x = _________________
3. x2
- 3x = 5 4.
x2
= 7 + 5x
Set = 0: ___________________= 0 Set = 0:
_____________________ = 0
a=____ b=____ c=____ a=____
b=____ c=____
Formula:
Formula:
Substitute:
Substitute:
449
5. x2
+ 3x - 4 = 0 6. x2
- 5x - 50 = 0
a=____ b=____ c=____ a=____
b=____ c=____
7.
x2
- 4x = 21 8. x2
+ 50 = 12x + 30
Set =
0: _______________= 0
_____________________ = 0
a=____ b=____ c=____ a=____
b=____ c=____
9. x2
- 5x = 0
(No constant term!) 10. x2
- 25 = 0
(No x term!)
a=____ b=____ c=____ a=____
b=____ c=____
450
EXAMPLE
3.
Solve for x: x2
+ 6x - 2 = 0 EXAMPLE 4.
Solve for x: x2
+ 25 = 6x
Solution:
Equation is already set = 0 Set equal
to zero: x2 - 6x + 25 = 0
a = 1 b = 6 c =
-2 a = 1 b = -6 c = 25
Negative
in the radical!!
No
Real Solution!
Notice that
in Example 3 the radical needed to be simplified, and then the fraction
reduced. Pay attention to these steps--traditionally they are difficult
for students and a predictable source of errors. Notice that in Example
4, there was a negative in the radical. This negative in the radical
means that there were no real solutions.
EXERCISES.
Solve the quadratic equations.
11. x2
- 4x - 6 = 0 12. x2
+ 4x + 2 = 0
451
13. x2
- 6x - 6 = 0 14. x2
= 4x + 8
15. 2x(x - 4) =
-7 16. 2x(x - 4) =
7
17. 3x2
+ 2(3 + x) = 4 - 6x
18. 3x(x - 4) = 7 - 8x
452
ANSWERS 5.07
p. 449 - 452:
1. 
;
2. 
;
3. 
;
4. 
;
5. -4, 1; 6.
10, -5; 7.
7, -3;
8. 10,
2; 9. 0, 5; 10. 5, -5; 11.
;
12.
;
13.
;
14. 
;
15.
;
16. 
;
17.
;
18. 7/3,-1.
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