Solving Quadratic Equations
An Example Of A Math Lesson In English 
(A
quadratic equation is a polynomial equation of degree two. The
standard form
is ax^{2} + bx + c = 0.)


Ok.. 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). 
