Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

2.4: Quadratic Inequalities

( \newcommand{\kernel}{\mathrm{null}\,}\)

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=b24ac.

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

x2x6>0

Solution:

First we solve the equality by factoring:

(x3)(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=x2x6

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.


This page titled 2.4: Quadratic Inequalities is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

  • Was this article helpful?

Support Center

How can we help?