Quadratic Equations
An example of a Quadratic Equation:
The function makes nice curves like this one:
Name
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:
2x^{2} + 5x + 3 = 0  In this one a=2, b=5 and c=3  
x^{2} − 3x = 0  This one is a little more tricky:


5x − 3 = 0  Oops! This one is not a quadratic equation: it is missing x^{2}
(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
ax^{2} + bx + c = 0
But sometimes a quadratic equation doesn't look like that!
For example:
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:
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 = \frac{−b + √(b^{2 }− 4ac)}{2a}
x = \frac{−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 xaxis.
 Or imagine the curve is so high it doesn't even cross the xaxis!
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:
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
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 bsquared 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 = \frac{−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
= −16
√(−16)
= 4i
(where i is the imaginary number √−1)
Answer: x = −0.2 ± 0.4i
The graph does not cross the xaxis. 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.
Example: Solve x^{2} − 4x + 6.25 = 0
= −9
√(−9) = 3i
(where i is the imaginary number √−1)
Answer: x = 2 ± 1.5i
The graph does not cross the xaxis. That is why we ended up with complex numbers.
BUT an upsidedown mirror image of our equation does cross the xaxis 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 = \frac{−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