Quadratics are ubiquitous in mathematics. For the purposes of this piece a quadratic is a real-valued function of the form

,

where such that . There is a little bit more to be said — particularly about the differences between a quadratic and a quadratic *function *but for those this piece is addressed to (third level: non-maths; all second level), the distinction is unimportant.

## Geometry

The basic object we study is the square function, , :

All quadratics look similar to . If then the quadratic has this geometry. Otherwise it looks like and has geometry

The geometry dictates that quadratics can have either zero, one or two real roots. A root of a function is an input such that . As the graph of a function is of the form , roots are such that , that is where the graph cuts the -axis. With the geometry of quadratics they can cut the -axis no times, once (like ), or twice.

Finding the roots of quadratic functions is a problem that occurs very often in mathematics. It is equivalent to solving a quadratic equation:

.

Trying to solve quadratic equations in your head is fraught with problems, and trying to do so leaves one with the impression that:

Finding when a Sum is Zero is HARD

The key to solving quadratics lies in this realisation along with the principle that:

Finding when a Product is Zero is EASY

Products are numbers multiplied together: is a sum, while is a product. This principle is a direct result of the following:

### No Zero Divisors

If and are two numbers such that , then or .

Therefore if a quadratic can be writen as a product:

,

then the roots of are given by the (hopefully) easier and . So how do we do this?

## Factorisation

Find a re-writing of as

such that (so as not to change the quadratic) and . From here, factorisation is inevitable using the distributive law:

.

### Example

Factor . Hence find the roots of .

*Solution: *We calculate . Now, from one, let us list the factors of 42:

.

We don’t have to go to seven as we already have seven. Now we need a pair of numbers with product *minus* so one of these has to negative. We also need the pair to add up to *minus … $latex 2\times*(-21)$ does the trick:

.

Now we take something common out of the first two terms and do the same for the second two terms in such a way that we are *left with the same factor twice:*

.

Now we can take out the common factor of :

.

Now once we have the factorisation, finding the roots is easy:

or

or

or .

### Example

Solve .

*Solution: *We calculate . Now, from one, let us list the factors of 15:

.

Now we need a pair of numbers with product *minus* so one of these has to negative. We also need the pair to add up to *… *this is impossible — we can only make the positive numbers 14 and two… factorisation has failed.

It is possible to solve the equation. It involves some gymnastics and writing it in the form:

.

These gymnastics can be undertaken for any quadratic and this leads us to a formula that can solve all quadratics:

## Formula

Suppose that is a quadratic. Then the roots of are given by:

.

### Example

Find the roots of .

*Solution*: Using the formula with , we have roots

.

## Appendices: Proofs

### The Geometry of the Graph

Of course, we have and . We will now compare the geometry of the graph of with the line . If the scales on the – and – axes are the same, is a line through the origin at an angle of .

Take now , then

for .

Consider . Then

.

This gives for .

We can also show to other facts. The slope near is small in the sense that if we take the two points on the graph of given by and (for but ), then the slope between them is given by:

.

We can also show that the slope between points is increasing from zero… this also shows that the function is increasing. Take so that the slope between and is given by:

which increases with .

Putting these facts together, goes through the origin and , and with an increasing slope, goes from ‘below’ for to ‘above’ for . Finally, the graph is symmetric about the -axis as

.

### All Quadratics are Translations of

Three facts that are presented without proof:

- The graph of is a horizontal translation of the graph of .
- The graph of is related to the graph of :
- if the the graph of is a horizontal scaling of ,
- if the graph of is an upside-down horizontal scaling of .

- The graph of is vertical translation of the graph of .

By multiplying out, any quadratic is of the form:

### Quadratic Formula

All quadratic equations of the form can be written in the form:

.

(To see this is equivalent to the original, multiply out and divide by ). Solve this to get the ‘‘ formula.

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