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:
 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 graph it makes, and
 the solutions (called "roots").
Hidden Quadratic Equations!
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 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:
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 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:
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 Complex solutions ... what does that mean?
It means our answer will include Imaginary Numbers. Wow!
Example: Solve 5x² + 2x + 1 = 0
√(−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.
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