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Mathematics LibreTexts

2.4: Quadratic Inequalities

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

 

Example 1

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 2

Solve

\[x^2- x - 6 > 0\]

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

Example 3

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:

\[\begin{align} \text{Area of walk} &= p(4+r)^2-p(r)^2 \\ &= .44(p)(r)^2 \end{align}\]

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

\[r = 1.1 \;\;\; \text{or} \;\;\; r = -.1.\]

since -.1 does not make sense, we can say that the radius of the garden is 1.1 feet.

Exercise

The profit function for burgers at Heavenly is given by

\[P = 35x - \dfrac{x^2}{25,000,000} - 40,000.\]

Where \(x\) represents the number of skiers that come on a given day. How many skiers paying for Heavenly will produce the maximal profit?

Contributors

  • Integrated by Justin Marshall.