A quadratic equation is an equation where the power of the variable is at most
2. The following are examples of quadratic equations.
Quadratic equations differ from linear equations by the fact that a linear
equation only has one solution, while a quadratic equation has
at most two solutions. There are some special situations when a quadratic equation only
has one solution.
We solve quadratic equations by factorisation, that is writing the quadratic as
a product of two expressions in brackets. For example, we know that:
In order to solve:
we need to be able to write
as
, which we already know
how to do.
Investigation : factorising a quadratic
Factorise the following
quadratic expressions:
Being able to factorise a quadratic means that you are one step away from
solving a quadratic equation. For example,
can be written as
. This means that both
and
, which gives
and
as the two solutions to the quadratic equation
.
Method: solving quadratic equations
First divide the entire equation by any common factor of the coefficients,
so as to obtain an equation of the form
where
,
and
have no common factors. For example,
can be written as
by dividing by 2.
Write
in terms of its factors
.
This means
.
Once writing the equation in the form
, it then
follows that the two solutions are
or
.
For each solution substitute the value into the original equation to check whether it is valid
Solutions of quadratic equations
There are two solutions to a
quadratic equation, because any
one of the values can solve the
equation.
Solve for
:
As we have seen the factors of
are
and
.
We have
or
Therefore,
or
.
Text here
for
or
.
Sometimes an equation might not look like a quadratic at first glance but
turns into one with a simple operation or two. Remember that you have to do the same operation on both sides of the equation for it to remain true.
You might need to do one (or a combination) of:
For example,
This is raising both sides to the power of
. For example,
This is raising both sides to the power of 2.
For example,
You can combine these in many ways and so the best way to develop your intuition for the best thing to do is practice problems. A combined set of operations could be, for example,
Solve for
:
Both sides of the equation should be squared to remove the square root sign.
The factors of
are
.
We have
or
Therefore,
or
.
Substitute
into the original equation
:
Therefore LHS
RHS. The sides of an equation must always balance, a potential solution that does not balance the equation is not valid. In this case the equation does not balance.
Therefore
.
Now substitute
into original equation
:
Therefore LHS = RHS
Therefore
is the only valid solution
for
only.
Solve the equation:
.
The equation is in the required form, with
.
You need the factors of 1 and 4 so that the middle term is
So the factors are:
Therefore
or
.
Therefore the solutions are
or
.
Find the roots of the quadratic
equation
.
There is a common factor: -2.
Therefore, divide both sides of the equation by -2.
The middle term is negative. Therefore, the factors are
If we multiply out
, we get
.
In this case, the quadratic is a perfect square, so there is only one solution
for
:
.