Quadratic Equations

An example of a Quadratic Equation:

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

The function can make nice curves like this one:

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:

a, b and c are known values. a can't be 0
x is the variable or unknown (we don't know it yet)

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, as 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)

Have a Play With It

Play with the Quadratic Equation Explorer so you can see:

the function's graph, and
the solutions (called "roots").

Hidden Quadratic Equations!

As we saw before, the Standard Form of a Quadratic Equation is

ax² + bx + c = 0

But sometimes a quadratic equation does not 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 Them?

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

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

Quadratic Graph

There are usually 2 solutions (as shown in this graph).

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 an example with two answers:

Quadratic Graph

But it does not always work out like that!

Imagine if the curve "just touches" the x-axis.
Or imagine the curve is so high it doesn't even cross the x-axis!

This is where the "Discriminant" helps us ...

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:

when b² − 4ac is positive, we get two Real solutions
when it is zero we get just ONE real solution (both answers are the same)
when it is negative we get a pair of Complex solutions

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
And we see them on this graph.

Answer: x = −0.2 or x = −1

Let's check the answers:

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":

♫ All around the mulberry bush
The monkey chased the weasel
The monkey thought 'twas all in fun
Pop! goes the weasel ♫

♫ x is equal to minus b
plus or minus the square root
of b-squared minus four a c
ALL over two a ♫

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 a pair of 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 a way it is easier: we don't need more calculation, we leave it as −0.2 ± 0.4i.

Example: Solve x² − 4x + 6.25 = 0

Coefficients are:a=1, b=−4, c=6.25

Note that the Discriminant is negative:b² − 4ac = (−4)² − 4×1×6.25
= −9

Use the Quadratic Formula:x = −(−4) ± √(−9) 2

√(−9) = 3i
(where i is the imaginary number √−1)

So:x = 4 ± 3i 2

Quadratic Graph with Cmplex Roots

Answer: x = 2 ± 1.5i