Solving Quadratic Inequalities

... and more ...

Quadratic

A Quadratic Equation (in Standard Form) looks like:

Quadratic Equation
A Quadratic Equation in Standard Form
(a, b, and c can have any value, except that a can't be 0.)

That is an equation (=) but sometimes we need to solve inequalities like these:

Symbol
 
Words
 
Example
>
 
greater than
 
x2 + 3x > 2
<
 
less than
 
7x2 < 28
 
greater than or equal to
 
5 ≥ x2 − x
 
less than or equal to
 
2y2 + 1 ≤ 7y

Solving

Solving inequalities is very like solving equations ... we do most of the same things.

When solving equations we try to find points,
such as the ones marked "=0"
Graph of Inequality
But when we solve inequalities we try to find interval(s),
such as the one marked "<0"

So this is what we do:

  • find the "=0" points
  • in between the "=0" points, are intervals that are either
    • greater than zero (>0), or
    • less than zero (<0)
  • then pick a test value to find out which it is (>0 or <0)

Here is an example:

Example: x2 − x − 6 < 0

x2 − x − 6 has these simple factors (because I wanted to make it easy!):

(x+2)(x−3) < 0

 

Firstly, let us find where it is equal to zero:

(x+2)(x−3) = 0

It is equal to zero when x = −2 or x = +3
because when x = −2, then (x+2) is zero
and when x = +3, then (x−3) is zero

 

So between −2 and +3, the function will either be

  • always greater than zero, or
  • always less than zero

We don't know which ... yet!

Let's pick a value in-between and test it:

At x=0:  x2 − x − 6  =  0 − 0 − 6  
−6

 

So between −2 and +3, the function is less than zero.

And that is the region we want, so...

x2 − x − 6 < 0 in the interval (−2, 3)

 

Note: x2 − x − 6 > 0 on the interval (−∞,−2) and (3, +∞)

 

And here is the plot of x2 − x − 6:

  • The equation equals zero at −2 and 3
  • The inequality "<0" is true between −2 and 3.
  x^2-x-6

 

 

What If It Doesn't Go Through Zero?

x^2-x-1

Here is the plot of x2 − x + 1

There are no "=0" points!

But that makes things easier!

Because the line does not cross through y=0, it must be either:

  • always > 0, or
  • always < 0

So all we have to do is test one value (say x=0) to see if it is above or below.

 

A "Real World" Example

A stuntman will jump off a 20 m building.

A high-speed camera is ready to film him between 15 m and 10 m above the ground.

When should the camera film him?

We can use this formula for distance and time:

d = 20 − 5t2

  • d = distance above ground (m), and
  • t = time from jump (seconds)

(Note: if you are curious about the formula, it is simplified from d = d0 + v0t + ½a0t2 , where d0=20, v0=0, and a0=−9.81, the acceleration due to gravity.)

OK, let's go.

 

First, let us sketch the question:

Jump Sketch

The distance we want is from 10 m to 15 m:

10 < d < 15

And we know the formula for d:

10 < 20 − 5t2 < 15

 

Now let's solve it!

First, let's subtract 20 from both sides:

−10 < −5t2 <−5

 

Now multiply both sides by −(1/5). But because we are multiplying by a negative number, the inequalities will change direction ... read Solving Inequalities to see why.

2 > t2 > 1

 

To be neat, the smaller number should be on the left, and the larger on the right. So let's swap them over (and make sure the inequalities still point correctly):

1 < t2 < 2

 

Lastly, we can safely take square roots, since all values are greater then zero:

√1 < t < √2

We can tell the film crew:

"Film from 1.0 to 1.4 seconds after jumping"

Higher Than Quadratic

The same ideas can help us solve more complicated inequalities:

Example: x3 + 4 ≥ 3x2 + x

First, let's put it in standard form:

x3 − 3x2 − x + 4 ≥ 0

This is a cubic equation (the highest exponent is a cube, i.e. x3), and is hard to solve, so let us graph it instead:

Graph of Inequality

The zero points are approximately:

  • −1.1
  • 1.3
  • 2.9

And from the graph we can see the intervals where it is greater than (or equal to) zero:

  • From −1.1 to 1.3, and
  • From 2.9 on

In interval notation we can write:

Approximately: [−1.1, 1.3] U [2.9, +∞)