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(male narrator) In this video,
we will look at solving polynomials
that have x squared in them for x.
Whenever there is an x squared in an x equation,
we need to factor the equation...
before we can solve using the zero product property.
If the factors both...if the factors are equal to 0,
then one of the factors must be 0.
Considering this equation...
before we can solve it for x, we first must factor it.
Using the AC method,
we wanna multiply to -12 and add to -4.
A little bit of work tells us that it's 6 and 2,
if the 6 is negative.
Because we have a 1 in front of x squared,
we can use those numbers in our factors:
x minus 6, times x, plus 2, equals 0.
Now that we have it factored,
we know one of those factors must be the 0,
so we set up two equations-- one for each factor.
Either the x minus 6 equals 0, or the x plus 2 equals 0.
We can quickly solve these equations
by adding 6 to the first...
to get x equals 6,
and subtracting 2 from the other
to get x equals -2.
These are the two numbers
that when plugged in for x in the original equation
will give us 0 for our final answer.
Let's take a look at another problem
where we have to factor first
before we can find our solutions.
In this problem, again, you notice we have an x squared.
Whenever there's an x squared,
we must factor in order to find the solutions.
We'll factor using the AC method,
multiplying to -12 and adding to +1.
This is 4 and -3.
That's how we're going to split up the x
into 4x and -3x.
We're ready to factor now using the reverse FOIL method.
The only way to multiply to 3x squared is 3x times x,
which means the outside had to be multiplied by 3
to give us one of our numbers.
Clearly, the only factor with a 3 in it is the -3x.
We know 3x times -1 would give us the -3x.
On the center, x times something
must give us the other number: the 4x.
Clearly, the way to get 4x from x is to multiply by 4.
Our expression is now factored.
Once it's factored, in order to find our solutions,
one of the factors must be equal to 0.
Either 3x plus 4 is 0, or the x minus 1 is 0.
We can solve by subtracting 4 from the first equation,
giving us 3x equals -4; finally, dividing by 3.
The first factor is 0 when x is -4/3.
To find our other possible answer for x,
we solve the other equation
by adding 1 to both sides to get x equals 1.
If x is either 1 or -4/3, the original equation,
when worked out, will equal 0.
Whenever there is an x squared in the problem,
we will always factor it
in order to solve using the zero product rule,
setting each factor equal to 0.