2.4: Quadratic Inequalities
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Solving Quadratic Equations
We solve quadratic equations by either factoring or using the quadratic formula.
Definition: 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 |
so that the quadratic has no real roots.
Quadratic Inequalities
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.
Applications
since -.1 does not make sense, we can say that the radius of the garden is 1.1 feet.
Where x represents the number of skiers that come on a given day. How many skiers paying for Heavenly will produce the maximal profit?
Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.