Quadratic Equations

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x 2 ).

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:

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

ax 2 + bx + c = 0

But sometimes a quadratic equation does not look like that!

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

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 use the special Quadratic Formula:

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 + √(b 2 − 4ac) 2a

x = −b − √(b 2 − 4ac) 2a

Here is an example with two answers:

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 b 2 − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

when b 2 − 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 5x 2 + 6x + 1 = 0

Coefficients are: a = 5, b = 6, c = 1 Quadratic Formula: x = −b ± √(b 2 − 4ac) 2a Put in a, b and c: x = −6 ± √(6 2 − 4×5×1) 2×5 Solve: x = −6 ± √(36− 20) 10 x = −6 ± √(16) 10 x = −6 ± 4 10 x = −0.2 or −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 ± √(b 2 − 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 b 2 − 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 5x 2 + 2x + 1 = 0

Coefficients are: a=5, b=2, c=1 Note that the Discriminant is negative: b 2 − 4ac = 2 2 − 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

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 x 2 − 4x + 6.25 = 0

Coefficients are: a=1, b=−4, c=6.25 Note that the Discriminant is negative: b 2 − 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

Answer: x = 2 ± 1.5i

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

BUT an upside-down mirror image of our equation does cross the x-axis at 2 ± 1.5 (note: missing the i).

Just an interesting fact for you!

Summary

Quadratic Equation in Standard Form: ax 2 + bx + c = 0
Quadratic Equations can be factored
Quadratic Formula: x = −b ± √(b 2 − 4ac)2a
When the Discriminant (b 2 −4ac) is:
- positive, there are 2 real solutions
- zero, there is one real solution
- negative, there are 2 complex solutions