# Quadratic equations

In first degree equations the maximum power any variable or
pronumeral is raised to is ‘1’. In *second degree* equations, the maximum
power that any variable or pronumeral is raised to is ‘2’. *Quadratic
equations* are second degree equations which *always* have a variable
raised to the power ‘2’.

_{} is a quadratic equation, because
‘x’ is a variable and it’s raised to the power ‘2’. It could also be written _{}.

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_{} is also a quadratic equation,
because ‘x’ is a variable, and it’s raised to the power ‘2’. This equation
however also has a term where ‘x’ is raised to the power ‘1’. It’s still a
quadratic equation though.

_{} is ** not** a quadratic
equation, because it has a variable raised to a power

*higher*than ‘2’.

There are many different types of quadratic equations. The simplest ones have only a term with the variable squared in it, and also another number term, like this:

_{}

Now, it’s pretty simple to solve this, we just need to take the square root of both sides:

_{}

However, there are actually *always* *two*
answers to a quadratic equation. We’ve just found one solution. But what
about if x was equal to *negative three*?

If
_{} then:

_{}

So if you put a value of _{} into the quadratic equation, it *also*
makes it true. This is because when you multiply two *negative* numbers
together, you always get a *positive* number.

So when you get to the *square rooting* stage in a
quadratic equation problem, you can use a _{} symbol in front of the square root
sign to indicate that you’re finding both the ‘plus’ and the ‘minus’ solution:

_{}

So now our answer is saying that x can equal ‘3’ ** or**
‘–3’.

### Only one answer to the square root of a number by itself

Be careful though, there is only *one* answer when you
take the square root of a number just by itself:

_{}

You only get the two ‘plus’ and ‘minus’ answers when you’ve got the square of a variable involved, and you need to take the square root of both sides, like this:

_{}

You can complicate this type of equation a little bit more if there is a coefficient in front of the squared variable term:

_{}

Then you need to divide both sides by the coefficient before you do the square root:

_{}

And you won’t always get a whole number answer, like in this case:

_{}

Sometimes you’ll end up trying to find the square root of a
*negative* number, like in this case:

_{}

Now, based on our current knowledge, there is no number
that you can multiply by itself to give a *negative* number. A positive
number multiplied by itself will give you a positive answer. But a negative
number multiplied by itself will also give you a positive answer. This means
that there is no way we can work out what the square root of a negative number
is. So there is no solution to this quadratic equation. In grades 11 and 12
in advanced mathematics you may cover a way to solve this equation, but no need
to worry about that now.

You could however simplify it a little bit if you wanted, remembering that you can split square roots up like this:

_{}

### Quadratics with a power 1 term but no number only term

A slightly more complicated quadratic equation is one where
you have both a *squared* term and a term where the variable is raised
just to the power 1, but no number only term, like this:

_{}

You can solve these types of quadratic equations by factorising them. For instance, in this equation all terms have a common factor of ‘x’, so we factorise it like this:

_{}

Now, the trick here is to remember that when you multiply *anything*
by *zero*, your answer is zero. This means that either the ‘x’ outside
the brackets is equal to 0, or the brackets term is equal to 0:

_{}

This is how we get our two answers, the answers that make these equations true. One answer is obviously x = 0. The other answer we can calculate:

_{}

So our overall answer is that x can equal either 0, or –7:

_{}

Sometimes you’ll get a question which is already factorised, like this one:

_{}

Because their product is equal to zero, one of the brackets
must equal zero, so we have *two* very simple equations to solve in order
to find the *two* possible values of a:

_{}