We solve quadratic equations by either factoring or using the quadratic formula.

 Definition of the Discriminant We define the discriminant  of the quadratic            ax2 + bx + c  as           D = b2 - 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

1045456564x2 + 3x + 2134534265256

have?

Solution

It is clear that 4ac is larger than b2 = 9.

Hence

D = 9 - 4ac < 0

So that the quadratic has no real roots.

Example:

Solve

x2 - 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 y-Value 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 = x2 - x - 6

stays above the x-axis when x < -2 and when x > 3.

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 - r2 = .44r2

multiplying out, we get,

16 + 8r + r2 -r2 = .44r2

or

.44r2 -8r -16

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 - x2/25,000,000 - 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?