Quadratic Equations

An example of a Quadratic Equation:

A Quadratic Equation 5x^2 - 3x + 3 = 0

Quadratic Equations make nice curves, like this one:

quadratic soccer kick

Name

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).

It is also called an "Equation of Degree 2" (because of the "2" on the x)

Standard Form

The Standard Form of a Quadratic Equation looks like this:

Quadratic Equation: ax^2 + bx + c = 0

 

Here are some examples:

2x2 + 5x + 3 = 0   In this one a=2, b=5 and c=3
     
x2 − 3x = 0   This one is a little more tricky:
  • Where is a? Well a=1, and we don't usually write "1x2"
  • b = −3
  • And where is c? Well c=0, so is not shown.
5x − 3 = 0   Oops! This one is not a quadratic equation: it is missing x2
(in other words a=0, which means it can't be quadratic)

 

Quadratic Graph

Have a Play With It

Play with the "Quadratic Equation Explorer" so you can see:

 

Hidden Quadratic Equations!

The Standard Form of a Quadratic Equation is

ax2 + bx + c = 0

But sometimes a quadratic equation doesn't look like that!

For example:

In disguise In Standard Form a, b and c
x2 = 3x − 1 Move all terms to left hand side x2 − 3x + 1 = 0 a=1, b=−3, c=1
2(w2 − 2w) = 5 Expand (undo the brackets),
and move 5 to left
2w2 − 4w − 5 = 0 a=2, b=−4, c=−5
z(z−1) = 3 Expand, and move 3 to left z2 − z − 3 = 0 a=1, b=−1, c=−3

 

How To Solve It?

The "solutions" to the Quadratic Equation are where it is equal to zero.

They are also called "roots", or sometimes "zeros"

There are usually 2 solutions (as shown in the graph a bit further up).

And there are a few different ways to find the solutions:

We can Factor the Quadratic (find what to multiply to make the Quadratic Equation)
Or we can Complete the Square
Or we can use the special Quadratic Formula:

Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a

Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula

Plus/Minus

First of all what is that plus/minus thing that looks like ± ?

The ± means there are TWO answers:

x = −b + √(b2 − 4ac) 2a

x = −b √(b2 − 4ac) 2a

Here is why we can get two answers:

Quadratic Graph

But sometimes we don't get two real answers, and the "Discriminant" shows why ...

Discriminant

Do you see b2 − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

Complex solutions? Let's talk about them after we see how to use the formula.

 

Using the Quadratic Formula

Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x2 + 6x + 1 = 0

Coefficients are:a = 5, b = 6, c = 1
Quadratic Formula:x = −b ± √(b2 − 4ac) 2a
Put in a, b and c:x = −6 ± √(62 − 4×5×1) 2×5
Solve:x = −6 ± √(36 − 20) 10
 x = −6 ± √(16) 10
 x = −6 ± 4 10
 x = −0.2 or −1

 

5x^2+6x+1

Answer: x = −0.2 or x = −1

 

And we see them on this graph.

Check -0.2:   5×(−0.2)² + 6×(−0.2) + 1
= 5×(0.04) + 6×(−0.2) + 1
= 0.2 − 1.2 + 1
= 0
Check -1:   5×(−1)² + 6×(−1) + 1
= 5×(1) + 6×(−1) + 1
= 5 − 6 + 1
= 0

 

Remembering The Formula

A kind reader suggested singing it to "Pop Goes the Weasel":

  "x is equal to minus b     "All around the mulberry bush
plus or minus the square root   The monkey chased the weasel
  of b-squared minus four a c     The monkey thought 'twas all in fun
  ALL over two a"     Pop! goes the weasel"

Try singing it a few times and it will get stuck in your head!

Or you can remember this story:

x = −b ± √(b2 − 4ac) 2a

"A negative boy was thinking yes or no about going to a party,
at the party he talked to a square boy but not to the 4 awesome chicks.
It was all over at 2 am.
"

Complex Solutions?

When the Discriminant (the value b2 − 4ac) is negative we get Complex solutions ... what does that mean?

It means our answer will include Imaginary Numbers. Wow!

Example: Solve 5x² + 2x + 1 = 0

Coefficients are:a = 5, b = 2, c = 1
Note that the Discriminant is negative:b2 − 4ac = 22 − 4×5×1 = −16
Use the Quadratic Formula:x = −2 ± √(−16) 10

√(−16) = 4i
(where i is the imaginary number √−1)

So:x = −2 ± 4i 10

 

5x^2+6x+1

Answer: x = −0.2 ± 0.4i

 

The graph does not cross the x-axis. That is why we ended up with complex numbers.

In some ways it is easier: we don't need more calculation, just leave it as −0.2 ± 0.4i.

 

Summary

 

 
(Hard Questions: 1 2 3 4 5 6 7 8 )