
(A quadratic
equation is a polynomial equation of degree two.
The standard form is ax^{2} + bx
+ c = 0.)
There's
no magic to solving quadratic equations. Quadratic
equations can be solved by
factoring and also by
graphing.

The
factoring method of
solution: 
Let's
do a quick review of factoring.
(If you need a more indepth look at factoring,
check the Factoring Section of this site.)
There
are primarily three types of factoring: 

*Common
Monomial 
ab
+ ac = a(b + c) 
*Difference
of Squares 
x^{2 } 9
= (x +3)(x  3) 
*Quadratic
Trinomial 
x^{2 } 5x
+ 6 = (x  3)(x  2) 
If you can factor, you
will be able to solve factorable quadratic equations.
Let's see how it is done.
Solve for x:
Here are the steps you should follow:
Solve
for x: x^{2 }+ 3x = 0
Factor
the common monomial. 
x(x
+ 3)=0 
Set
each factor equal to 0 and solve for x. 
x = 0 

x + 3 = 0
x = 3 
List
all values of x. 
x
= {0, 3} 
Solve
for y: y^{ 2} = 16
Get
all terms on the same side. 
y ^{2 } 16 = 0 
Factor
the difference of squares. 
(y + 4)(y
 4) =0 
Set
each factor equal to 0 and solve for y. 
y + 4 = 0
y = 4 

y  4 = 0
y = 4 
List
all values of y. 
y
= {4, 4} 
Solve
for c: c^{ 2 } 12 = c
Get
all terms on the same side. 
c^{ 2 }
12  c =0 
Arrange
the terms in standard form. 
c ^{2 }
c  12 = 0 
Factor
the quadratic trinomial. 
(c + 3)(c
 4) = 0 
Set
each factor equal to 0 and solve for c. 
c + 3 = 0
c = 3 

c  4 = 0
c = 4 
List
all values of c. 
c
= {3, 4} 
Solve
for x:
Employ
"product of the means = product of the extremes" (crossmultiply)
for this proportion. 

Get
all terms on the same side. 
x ^{2 } 1296 = 0 
Factor
the difference of squares. 
(x +
36)(x
 36) =0 
Set
each factor equal to 0 and solve for x. 
x + 36 = 0
x = 36 

x  36 = 0
x = 36 
List
all values of x. 
x
= {36, 36} 
Solve
for x:
Write a quadratic
equation, in the form ax^{2 }+ bx + c
= 0, whose roots are 2 and 5.
The simplest
answer will be an equation where the factors of the expression are
(x  2) and (x  5). Create this equation. 
(x  2)(x  5) = 0 
Multiply. 
x ^{2 } 5x  2x + 10 = 0 
Combine to get an answer equation. 
x ^{2 }
7x + 10 = 0 
The square of a
number exceeds 5 times the number by 24. Find the number(s).
Translate the
problem into a mathematical equation. 
x^{2}
= 5x + 24 
Get
all terms on the same side. 
x ^{2 } 5x  24 = 0 
Factor
the difference of squares. 
(x
 8)(x +
3) =0 
Set
each factor equal to 0 and solve for x. 
x  8 = 0
x = 8 

x + 3 = 0
x = 3 
List
all values of x. 
x
= {8, 3} 
In football, the
height of the football reached during a pass can be modeled by the equation h = 16t ^{2} +
28t
+ 6, where the height, h, is in feet and the time, t, is
in seconds. How long does it take for this ball to reach a
height of 12 feet?
Substitute 12
into the equation for h. 
12 = 16t^{ 2} +
28t + 6 
Get
all terms on the same side. Move terms to the left side to
avoid working with a negative leading coefficient. 
16t^{ 2}  28t + 6 = 0 
Factor
the quadratic trinomial. 
(4t

1)(4t
 6) =0 
Set
each factor equal to 0 and solve for t. 
4t  1 = 0
4t = 1
t = 1/4


4t 
6 = 0
4t = 6
t = 6/4=3/2 
List
all values of t that are positive. Negative time,
should it appear, is not considered an answer. 
t
= {1/4, 3/2}. Reaches a
height of 12 feet when time is 0.25
seconds (ball going up) and 1.5 seconds (ball coming down). 

See how to use
your
TI83+/84+ graphing calculator with quadratic equations.
Click calculator. 

