In this video we're talking about how to use the Zero Theorem to find the solutions to a quadratic equation.
And in this particular example we have the equation negative x plus x squared is equal to 6.
Now before we talk about what the Zero Theorem allows us to do the first thing we know right away is
that we need to reorganize this equation because we should always write the terms of our equation in
descending order of degrees so the highest degree variable should go first which means this x squared
term should come before this x to the first term or just x.
So we should put x squared first x squared and then or minus X and then in order to use the Zero Theorem
the right hand side of the equation is always going to need to be equal to zero.
So in order to get the right hand side to be equal to zero we need to go ahead and subtract six from
both sides of the equation.
So then on the right here we get 6 minus 6 equals zero.
So just go ahead and put negative six over here and that's equal to zero the zero theorem is useful
because it allows us to find the solutions to quadratic equations which are different than linear equations
because quadratic equations have a second degree variable involved.
In other words they have an x squared term involved where linear equations just had an X to the first
term involved.
In other words we've been used to solving equations like X minus three is equal to six.
We would just add three to both sides and we would get x is equal to 9 and X equals 9 would be a solution
to this equation.
This is a linear equation because the only variable involved is just this x to the first variable.
But here we have an x squared variable which means we're dealing with a quadratic equation.
Obviously finding the solutions to this equation is going to be more complicated than finding the solution
to this linear equation here.
And that's where the zero theorem comes in the zero there and tells us that when we have any equation
p times Q equals zero.
When this is the case then we know that either p equals zero and or q is equal to zero.
And that doesn't make sense to us because if this is our equation here p times Q equals zero or just
p q equals zero.
The only way this is going to be true is if p is zero because then we would get 0 Q equals zero or just
zero equals zero and that would be a true equation or if Q is equal to zero because then we would get
zero P is equal to zero or just zero equals zero and that would also be true.
Or if both of them were equal to zero then we'd have zero time zero equals zero.
And that would also be a true equation.
But there's no way that this equation can be true if neither P is equal to zero nor Q is equal to zero.
At least one of them must be zero or both of them will be zero.
So this is the Zero Theorem and we can use it to find the solution to this equation.
And the way that we do that is by factoring the left hand side here.
This is just a regular trinomial that we've already learned how to factor.
And we're just going to factor it into two binomials that are multiplied together.
And when we do that we're going to get X and X and then we have factors of 6 that are 3 and 2.
So we can go ahead and say minus three and plus two.
And if we multiply this back out we get X squared plus 2 x minus 3 x would be minus 1 x and then negative
3 in positive 2 gives us negative sex so we factored this correctly.
And now all we need to realize is that here we have p times Q But basically now all we've done by factoring
is say that we have p times Q equals zero like this.
So our equation or quadratic equation is now in the same form as the equation in the Zero Theorem and
all we need to say is that in order for this equation to be true one or both of these factors must be
equal to zero.
So either X minus 3 must be equal to zero and or X plus two must be equal to zero.
And then we can solve each of these linear equations just like we solved this linear equation up here.
We're back to something that's really simple for us to do.
So this first one we can just add three to both sides and we get x is equal to positive 3.
Here we'll subtract two from both sides and get x is equal to negative 2.
What this tells us is that 3 and negative 2 are the solutions to our quadratic equation because of X
is 3.
Then this first factor will be 0 and will get 0 times quantity x plus 2 or just 0 equals zero and X
equals negative 2 is another solution to the equation.
So now what this tells us is that X equals 3 and X equals negative 2 are both solutions to the equation
and we can prove it to ourselves by plugging these back in.
So if I say that X equals three is a solution to the equation and I plug it back into my factoring equation
here what I'm going to get is three minus three.
In this first vector here instead of X minus three three minus three and then three plus two equals
zero.
And when I simplify I'll get 0 times 5 is equal to zero zero times 5 is just zero.
So I get zero equals zero.
And that checks out the other solution is exit equals negative 2.
And if I plug that in I'm going to get negative two minus three negative two plus two equals zero and
when I simplify negative to minus three gives me negative 5 negative 2 plus 2 gives me zero.
And I know that negative five times zero is zero.
So I get zero equals zero and that checks out also.
So I can say that X equals three and X equals negative two are solutions to the equation.
And that's how you use the zero theorem to find solutions to a quadratic equation.