Finally our last method for solving quadratic equations is by using the quadratic formula.
So we're actually going to solve this problem using the same
quadratic equation that we had with our completing the square method.
This way you can see how the quadratic formula works,
and that we do get an equivalent answer to what we had with completing the square when using the quadratic formula.
We can always use the quadratic formula as long as we have a quadratic equation.
So in any form so even when we went back to the square root property and we had that 9x squared plus 25 equals 0,
we can still use the quadratic formula.
When we had
the square root property where we had a quantity in parentheses squared, we can still use the quadratic formula.
We would just have to simplify that quantity squared first and actually multiply that out. So we can always use this this.
Usually is the go-to for people because we know that it always works.
But factoring is the fastest method if we can factor. In this case x squared minus 6x plus 4 ,we can't factor
so we have to use quadratic formula.
So for a quadratic equation in standard form ax squared plus bx plus c equals 0 the solution to the quadratic
equation can be given by the formula, and you are going to have to memorize this.
x equals
negative b plus
or minus the square root of b squared minus
4ac
all over
2a. So it's going to come back enough in your math career that you really need to memorize that formula.
That's what we're going to use in order to solve this x squared minus 6x plus 4 equals 0.
So we have x squared minus 6x plus 4 equals 0 so in order to use the quadratic formula
we need to identify what a b and c are.
So a will be your coefficient of x squared which in this case would just be 1, b would be our
coefficient of x which would just be negative 6 ,and then c would be our constant
which is just positive 4. So all we're doing is just straight plugging in. So I'm going to have here then negative b.
So that means I'm doing the opposite of whatever b
is so I'm going to write the the opposite of negative six.
So you can see that there is the negative from the negative in front. And then b was negative six plus or minus the
square root of (b squared, so negative six squared,
minus four times a which was one and then times  c which is 4) all over
(2 times a)
which is 1.
So just straight plugging the values in. Now we're just simplifying this so
from here the opposite of negative 6 would be positive 6 plus or minus the
square root of (negative 6 squared would be a positive 36
minus 4 times 1 times 4 would be a minus 16 )all
over (2 times 1 is 2).
Continuing underneath the square root. we would have 36 minus 16 which is 20
all over 2. From here
we do want to simplify our radical so off to the side the square root of 20
largest perfect square factor is 4 so we have square root of 4 square root of 5.
So we would just get 2 square root of 5. So that means here that I have 6
plus or minus square root of 20 now becomes (2
square root of 5) all over 2. And then finally we just want to simplify this by dividing with our denominator.
So we're going to see if our denominator goes into each term in the numerator.
So we're going to divide 2 into 6 and then 2 into 2 square root of 5.
So if you want, you can write it that
way. You can see how it is simplified with the 2 underneath each piece.
This is helpful.
Especially when you can't simplify. That way you can see the fraction that you're left with, but in this case
we can simplify. We get our final answer of 6 divided by 2 would be 3
plus or minus, here
we're only looking at the number in front of our radical and then the denominator, so 2 goes into 2 one time.
So these would just cancel out to be
3 plus or minus the square root of 5, and this is the answer that we got in the previous video.
So it shows that our quadratic formula also worked for a problem where we needed to complete the square.
Again if we wanted to write this in
solution set notation,
we would have three plus the square root of 5 and then 3 minus the square root of 5.
That's how you use the quadratic formula.
