Quadratic Equations
An example of a Quadratic Equation:
Quadratic Equations make 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:

Here are some more 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) 
Hidden Quadratic Equations!
So 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(z1) = 3  Expand, and move 3 to left  z^{2} – z – 3 = 0  a=1, b=1, c=3 
5 + 1/x – 1/x^{2} = 0  Multiply by x^{2}  5x^{2} + x – 1 = 0  a=5, b=1, c=1 
Have a Play With ItPlay with the “Quadratic Equation Explorer” so you can see:

How To Solve It?
The “solutions” to the Quadratic Equation are where it is equal to zero.
There are usually 2 solutions (as shown in the graph above).
They are also called “roots“, or sometimes “zeros”
There are 3 ways to find the solutions:
 We can Factor the Quadratic (find what to multiply to make the Quadratic Equation)
 We can Complete the Square, 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:
Here is why we can get two answers: 
But sometimes we don’t get two real answers, and the “Discriminant” shows why …
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 two 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 ± √(b^{2}4ac) ] / 2a  
Put in a, b and c:  x = [ 6 ± √(6^{2}4×5×1) ] / (2×5)  
Solve:  x = [ 6 ± √(3620) ]/10  
x = [ 6 ± √(16) ]/10  
x = ( 6 ± 4 )/10  
x = 0.2 or 1 
Answer: x = 0.2 or x = 1And 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
I don’t know of an easy way to remember the Quadratic Formula, but a kind reader suggested singing it to “Pop Goes the Weasel”:
♫  “x equals 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!
Complex Solutions?
When the Discriminant (the value b^{2} – 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:  b^{2} – 4ac = 2^{2} – 4×5×1 = 16  
Use the Quadratic Formula:  x = [ 2 ± √(16) ] / 10  
The square root of 16 is 4i (i is √1, read Imaginary Numbers to find out more) 

So:  x = ( 2 ± 4i )/10 
Answer: x = 0.2 ± 0.4iThe 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.
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