2.4: Quadratic Inequalities
- Page ID
- 233
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\[ax^2 + bx + c \]
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 |
so that the quadratic has no real roots.
Quadratic Inequalities
\[x^2- x - 6 > 0\]
Solution:
First we solve the equality by factoring:
\[(x - 3)(x + 2) = 0\]
hence
\[x = -2 \; \text{ or } \; x = 3.\]
Next we cut the number line into three regions:
\[x < -2, -2 < x < 3, \text{ and } x > 3.\]
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:
\[x < -2 \; \text{ or } \; x > 3.\]
We will see how to verify this on a graphing calculator by noticing that
\[y = x^2 - x - 6 \]
stays above the 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.