 Solving Quadratic Equations
We solve quadratic equations by either factoring or using the quadratic formula.
Definition of the Discriminant
We define the discriminant of the quadratic
ax^{2} + bx + c
as
D = b^{2}  4ac 
The discriminant is the number under the square root in the quadratic
formula. We immediately get
D 
# of Roots 
> 0 
2 
< 0 
0 
0 
1 
Example
How many roots does
1045456564x^{2} +
3x + 2134534265256
have?
Solution
It is clear that 4ac is larger than b^{2}
= 9.
Hence
D = 9 
4ac < 0
So that the quadratic has no real roots.

Quadratic Inequalities
Example:
Solve
x^{2} 
x  6 > 0
Solution:
First we solve the equality by factoring:
(x  3)(x + 2) = 0
Hence
x = 2 or
x = 3
Next we cut the number line into three regions:
x < 2,
2 <
x < 3, and x > 3
On the first region (test x = 3), the quadratic is positive, on the second
region (test x = 0) the quadratic is negative, and on the third region (test
x = 5) the quadratic is positive.
Region 
Test Value 
yValue 
Sign 
x < 2 
x = 3 
y = 6 
+ 
2 < x
< 3 
x = 0 
y = 6 
 
x > 3 
x = 5 
y = 14 
+ 
We are after the positive values since the equation is "> 0".Hence our solution is region 1 and
region 2.
x < 2 or
x > 3
We will see how to verify this on a graphing calculator by noticing that
y = x^{2} 
x  6
stays above the xaxis when x < 2 and when
x > 3.

Applications
A 4 ft walkway surrounds a circular flower garden, as shown in the sketch.
The area of the walk is 44% of the area of the garden. Find the radius of
the garden.
Solution:
Area of the walk =
p(4 + r)^{2}  p( r)^{2}
= .44(p)( r)^{2}
Dividing by p we have,
(4 + r)^{2 }
r^{2} = .44r^{2}
multiplying out, we
get,
16 + 8r +
r^{2} r^{2} = .44r^{2}
or
.44r^{2}
8r 16
Now use the quadratic formula:
a = .44,
b = 8, c = 16
so
r = 1.1 or
r = .1
since .1 does not make sense, we can say that the radius of the garden is
1.1feet.
Example:
The profit function for burgers at Heavenly is given by
P = 35x 
x^{2}/25,000,000  40,000.
Where x represents the number of skiers that come on a given day. How many
skiers paying for Heavenly will produce the maximal profit?