Today we're going to be looking at
solving quadratic equations by using the
quadratic formula. First of all we need
to look at the quadratic formula and if
we write our equation in the form of ax
squared plus bx plus c. Then we know a, b,
c and we can plug it into our formula.
Which is x equals a negative b plus or
minus the square root of b squared minus
4ac all divided by 2a. This b squared
minus 4ac later on we will talk about
that is the part of the quadratic
equation that we call the discriminant.
First of all our first example is 8x
squared plus 11x plus 2 equals 0. So we
just plug our values in the quadratic
equation and when we do the math, we end
up with minus 10 plus or minus the
square root of 100 minus 64 divided by
16. Continuing to simplify we get a
negative 10 plus or minus the square
root of 36 all divided by 16 Then we
have a negative 10 plus or minus 6
divided by 16. Continuing to go ahead and
simplify, a negative 10 plus minus 4 in
other words we're going the top route, is
a negative 4 divided by 16 which reduces
to 1 negative 1 over 4. If we go the
bottom route which is our
negative 16 minus 6, we get a negative 16
over 16 which is a negative 1. Our second
example then is to solve x squared minus
2x equals 8. So we subtract 8 to both
sides and we end up with x squared minus
2x minus 8 equals 0. Again plugging it
into the quadratic equation we have our
2 plus or minus the square root of 4
plus 32
it's all divided by two. Then we go to
continue simplifying and we have 2 plus
or minus the square root of 36 equaling
two all divided, I'm sorry, all divided by
two. So we know the square root of 36 is
6. So now we're going to go to the top
route 2 plus 6 divided by 2 which is 4
or the bottom route which is 2 minus 6
divided by 2 which is a negative 2.
Continuing on we have 1/8 a squared plus
a plus 1/4 equals 0. Well I don't want to
work with fractions so I'm going to
multiply by the LCD which is 8 and when
I do that I get a squared plus 8 a plus
2 equals 0. Plugging that in our
quadratic formula and then we continue
to simplify under the radical I have 1
minus 8 which is a negative 7. We know
when we pull the negative out that gives
us i, so our final answer ends up being a
negative 8 plus or minus i times the
radical 7 all divided by 2. This is a
discriminant that I mentioned earlier in
the video. If we do that b squared minus
4ac and it's positive there are two real
solutions, if it equals zero there is one
real solution and if it's negative there
are two complex but not real solutions.
Continuing on we're going to use the
discriminant our first example is m
squared plus 6m plus 9 equals 0. Then we
give we have 6 squared minus 4 times 1
times 9 and that does equal 0. Then the
equation has one real solution doing the
same thing 4x squared plus 5 plugging it
in we have 0 squared minus 4 times 1
times 5 and that ends up giving us a
negative 20. Well then the equation has
too complex but not real solutions. And
then our final example on this page is
3x squared minus 5x minus 1 equals 0.
We plug that in the discriminant formula
we end up with a negative 5 squared
minus 4 times 3 times a negative 1 and
that's 37, well then the equation has two
real solutions.
Another problem is a rock is thrown
upward from the top of a 300 foot
building with a velocity of 10 feet per
second the height above ground h in feet
of the object after t seconds is given
as h equals a negative 16t squared plus
10t plus 300. So that were asked how long
will it take to strike the ground? Well
what is the height at the ground, it is
zero. So we set h equal to zero and go
ahead and solve using our quadratic
equation. We plug those in and we end up
getting a negative 10 plus or minus the
square root of 19,300 all divided by
negative 32. Sso when we do the math we
end up getting four point seven seconds.
