Quadratic Equations | Factoring (Level 2)
In the previous video we introduced a general procedure to solve quadratic equations by
factoring. As a refresher we will review the procedure, it consisted of 3 steps, the first
step was to write the quadratic equation with 0 on one side. The second step was to factor
the other side of the equation, and the final step was to solve the equation by setting
each linear factor equal to 0. Recall that in the previous video we practiced step 3,
specifically using the zero product property to find the zeros or roots of the quadratic
equation. Applying this last step is actually pretty easy granted that we are able to factor
the quadratic equation into linear factors. So how exactly do we factor a quadratic equation?
Well it depends on the type of quadratic equation you are faced with. In general there are two
types of quadratic equations that can be solved by factoring; they include quadratic binomials
of the form ax squared plus bx equals 0 and quadratic trinomials of the form ax squared
plus bx plus c equals 0. Let’s first go over quadratic binomials of the form ax squared
plus bx equals 0, quadratic equations of this form are pretty much guaranteed to be solvable
by factoring. This is because a quadratic binomial of this form always contains a variable
as a greatest common factor also known as the GCF and therefore can be factored out
every single time. Once we have it in factored form we go ahead and apply the zero product
property, notice that we will always obtain x = 0 as one solution when dealing with quadratic
binomials of the form ax squared plus bx equals 0. The second solution will be an expression
in terms of the coefficients a and b. Alright let's go ahead and try some examples.
Solve the equation x squared plus x equals 0.
First notice that we are dealing with a quadratic equation because the highest power out of
all the terms is 2 in other words we are dealing with a polynomial of degree 2. In addition,
notice that it’s a quadratic binomial of the form ax squared plus bx equals 0 where
both a and b are equal to 1. Keep in mind that quadratic equations of this form can
always be solved by factoring so we go ahead and solve this equation by using this technique.
The first step is to make sure that we have 0 on one side of the equation, we see that
this is the case it has been taking care of so the first step is done, the second step is
to factor the quadratic equation, we do this by noticing that each term on the left hand
side contains the variable x as its GCF or greatest common factor so we go ahead and
factor that out and multiply it by what is left after factoring in this case we leave
behind the quantity x + 1. So now we have the expression x times the quantity x + 1 equals 0.
Its always a good idea to check and make sure you factored the quadratic equation correctly
by distributing the factor and making sure you obtain the original quadratic equation.
Doing that we see that we indeed factored the expression the correct way. By factoring
the quadratic equation we have successfully completed step two, now the last step is to
apply the zero product property and set each factor equal to zero, recall that in the previous
video we went over various examples in carrying out this last step so at this point completing
this step should be more or less second nature for you. So we go ahead and set each factor
equal to zero. The first factor is automatically equal to zero, Setting the second factor equal
to zero and solving for x by subtracting 1 from both sides we obtain the second solution
x = -1, If we were to graph the left hand side of the equation we would obtain the following
graph, notice that this graph crosses the x-axis at (0, 0) and (-1, 0) these are the
roots or zeros of the quadratic equation. I want you to notice that when you solve quadratic
equations of form ax squared plus bx equals 0 you always obtain x = 0 as one solution
because this particular type of quadratic binomial always contains a single variable
as the GCF, which can be factored out every single time. Alright let’s go ahead and try the
next example.
Solve the equation x squared minus 5x equals 0.
Notice that we are dealing with a quadratic binomial for which both terms contain a single
variable in common. Before we factor anything out we need to make sure that one side is
set to zero, for this particular problem this has already been done for us so step 1 is complete,
second step is to factor the quadratic equation, so we do this by noticing that each
term contains a single variable in common in other words its GCF, greatest common factor
is x so we go ahead and factor this expression out and we are left with the quantity x minus 5,
having factored the quadratic equation we go ahead and apply the zero product property
to each of the two factors. Once again notice that one solution is equal to zero and for
the second solution we go ahead and solve the equation x minus 5 equals zero so we do
this my adding 5 to both sides of the equation, and obtain x = 5 as the second solution. Alright
let’s try last example.
Solve the equation 3x squared plus 2x equals 0.
Once again we are dealing with a quadratic binomial so we first need to set one side
equal to zero, and this has been taken care for us. Next we go ahead and factor the quadratic
equation, notice that both terms have a single variable in common so we go ahead and factor
that out, doing that we obtain x times the quantity 3x + 2 equals zero. Always check
to make sure you factored the expression correctly by mentally distributing what you initially
factored out and making sure you obtain the original quadratic equation. Having successfully
factored out the quadratic equation we go ahead and solve for x by applying the zero
product property. So we set each factor equal to zero. Doing that we obtain x = 0 as one
solution. The second solution is obtained by solving the equation 3x + 2 = 0 so we first
subtract 2 from both sides of the equation and obtain the following expression, next
we divide both sides by 3 and simplify, doing that we obtain x = negative 2/3 as the second
solution. Notice again that we always obtain x = 0 as one solution when dealing with these
types of quadratic binomials. Alright in our next video we will continue solving quadratic
binomials of the form ax squared plus bx equals 0 with slightly harder examples.