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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.

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

$.44r^2-8r -16.$

$a = .44, b = -8, c = -16$

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.