# 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:- Where is
**a**? Well**a=1**, as we don't usually write "1x^{2}" **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 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!

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:

**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 = \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 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:

*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

**or**−1

**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 b-squared 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

**Coefficients**are

**:**a=5, b=2, c=1

**Discriminant**is negative:b

^{2}− 4ac = 2

^{2}− 4×5×1

=

**−16**

**Quadratic Formula:**x = \frac{−2 ± √(−16)}{10}

*√(−16)* = 4**i**

(where **i** is the imaginary number √−1)

**Answer:** x = −0.2 ± 0.4**i**

The graph does not cross the x-axis. 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 ± 0.4**i**.

### Example: Solve x^{2} − 4x + 6.25 = 0

**Coefficients**are

**:**a=1, b=−4, c=6.25

**Discriminant**is negative:b

^{2}− 4ac = (−4)

^{2}− 4×1×6.25

=

**−9**

**Quadratic Formula:**x = \frac{−(−4) ± √(−9)}{2}

*√(−9)* = 3**i**

(where **i** is the imaginary number √−1)

**Answer:** x = 2 ± 1.5**i**

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 = \frac{−b ± √(b^{2 }− 4ac)}{2a}
- When the Discriminant (
**b**) is:^{2}−4ac- positive, there are 2 real solutions
- zero, there is one real solution
- negative, there are 2 complex solutions