2.4: Quadratic Inequalities

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Oct 6, 2021
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- 2.3: Curve Intersection
- Back Matter

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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

$ax^2 + bx + c$

$D = b^2 - 4ac.$

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.

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Solving Quadratic Equations

Quadratic Inequalities

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