- [Voiceover] Welcome to an example
on how to solve a quadratic equation
using the technique of factor by grouping.
Our goal here is to solve the equation
12x squared plus 20x equals -3.
We first wanna set this
equation equal to zero
so let's add three to both
sides of the equation.
This gives us the equation
12x squared plus 20x
plus three equals zero.
Now for our next step
we wanna factor the left
side of the equation.
To do this we'll use the
technique of factor by grouping
where the steps are shown below.
So for review, to factor
a quadratic expression
in the form ax squared
plus bx plus c by grouping,
we follow these five steps.
Where the first step is
to find the factors of ac
that add to b, let's begin
by identifying the values
of a, b, and c.
a is the coefficient of
x squared which is 12,
b is equal to the
coefficient of x which is 20,
and c is equal to the
constant term positive three.
So it is important
to make sure the equation is equal to zero
before we find the values of a, b, and c.
Now we wanna find ac which means a times c
which would be 12 times three
which equals positive 36.
So we wanna find the
factors of positive 36
that add to b, which equals 20.
So we're looking for two numbers
that when we multiply
them we get positive 36
but when we add them we get positive 20.
So if you can just think of
the two factors that we need
that's great, if not
I would recommend listing
all the factors of 36
which would be one times 36, two times 18,
three times 12, four times
nine, and six times six.
So we're looking for the
factors that have a sum of 20.
Notice how those factors
would be positive two
and positive 18.
Positive two and positive 18 multiply
and give us positive 36 and two plus 18
is equal to 20, which equals b.
So for step two we're now
going to write the bx term
or in this case the 20x
term as a sum or difference
using the factors from step one.
Which means we're going to write 20x
as 2x plus 18x.
So on the left side we
would have 12x squared,
again plus 2x plus 18x
plus three equals zero.
Notice on the left side we do
have an equivalent expression
because 2x plus 18x is equal to 20x.
And you might be asking would it be okay
to write 18x plus 2x here?
And the answer is yes,
it would still work.
Step three we now divide or
group the polynomial into halves
and because we have four
terms, the first group
or first half is going
to be the first two terms
and the second group or
second half is going to be
the second two terms.
Step four, we wanna factor
out the greatest common factor
from the first half and second half.
So looking at just the first two terms
we wanna factor out the
greatest common factor,
which would be 2x.
So we'll factor out 2x, again
from just the first two terms
that'll leave us with 6x plus one.
And now we want to factor
the greatest common factor
out of the second two
terms so the second group.
The greatest common factor of
18x plus three would be three
so we'll factor out a positive three
so we'll write plus and then three.
If we factor out three
we're left with 6x plus one.
And this is still equal to zero.
Notice on the left these
two products do share
a common binomial factor of 6x plus one,
which brings us to the last
step of factor by grouping.
We wanna factor out the
common binomial factor,
which again in this case is
the quantity 6x plus one.
So if we factor out 6x plus one
notice how we'd be left with
the quantity 2x plus three,
which is our second factor.
So now we have the left side factored
and this is equal to zero.
Well if we have a product
that's equal to zero
either the first factor
or the second factor
must be equal to zero.
Which is a zero product property.
So to solve the equation
we now set each factor
equal to zero and solve for x.
So this product equals zero
if 6x plus one equals zero
or 2x plus three equals zero.
So here if we solve for x
we would first subtract one
giving us 6x equals -1
and then we would divide by six
so we have x equals -1/6
and here we would first
subtract three on both sides
giving us 2x equals -3 and
dividing both sides by two.
Notice here we have x equals -3/2.
So we have two solutions,
x equals -1/6 or x equals -3/2.
These are the two values of x
that would satisfy the given equation.
Now it's a good time to
verify these solutions
by substituting these two x values
into the original equation
to make sure the left side
is equal to negative three.
But let's do this on the calculator.
So for x equals -1/6 we'd
have 12 times -1/6 squared
plus 20 times -1/6.
And notice how this is equal to -3,
which verifies our first solution.
And now we'll check x equals -3/2.
So we'd have 12 times -3/2 squared
and then plus 20 times -3/2.
And once again notice
how this does equal -3
which verifies that our
two solutions are correct.
I hope you found this helpful.
