Good day students welcome to mathgotserved.com
in this clip were going to going over how
to use the quadratic formula the instructions
for the example that we're going to be sold
inverse follows we are to describe the nature
of the roots of the given quadratic equation
will going to describe the nature of the roots
of the given quadratic equation using the
discriminant okay using the discriminant and
then we are to solve the equation using the
quadratic formula okay then solve or find
the roots the roots are the solution to the
quadratic equation okay and find the roots
or think about it as a solution using the
quadratic formula okay 
alright so be example that we're going to
be looking that's today is the quadratic equation
3X square +7 equals negative 2X okay so the
first thing we're going to do is you will
going to use the discriminant's to describe
the nature of the roots will before we start
roots are real quick review on what's the
discriminant's rule is in determining the
nature of the roots okay so recall the discriminant
of the component of the quadratic formula
be specific it is erratic Okay so discriminant
is b squared minus four A C (b^2-4ac)-only
represented with Tom the trinomial symbol
okay so if you discriminant Delta is Om negative
then that means that you going to have to
distinct imaginary roots okay only talk about
imaginary roots are going to have that I component
there not if you are discriminant is zero
then you going to have one real roots okay
and that is also known as a double root if
you discriminant is positive you going to
have two distinct roots okay they can be father
separated into rational or irrational that
depends on if the discriminant is eat perfect
square or not okay so let's say that the discriminant
is positive and E perfect square give it's
a perfect square was going happen is that
the Om square root components will become
an integer so you going to have a rational
result so this case you going to have two
distinct real roots that are rational two
distinct real rational roots okay now what
if you have a scenario where your discriminant
is not a perfect square so you discriminant
is positive and not a perfect square in that
case you going to have to distinct  real
irrational roots okay think about the rational
numbers as numbers that you cannot represents
as the quotient of two integers for example
you have square roots something like this
roots to is irrational because the cannot
be represented by the quotient of two integers
arrive a number that can be represented of
a fraction of two integers is a irrational
number right so let's give this for cases
in mind and let's take a look at the quadratic
equation that we doing with happens going
to do in order to find the discriminant and
uses to this determine the nature of the roots
is we need to write down the discriminant
equation okay so part one is to find it discriminant
three right so our discriminant Delta is b
squared minus four A C (b^2-4ac) now we need
to find a BNC now one mistake the students
make is the filled to put the equation in
standard form before  finding a BNC okay
so the give ourselves a nice little caution
here place equation in standard form first
okay what is the standard form sum up form
is a X square plus be X plus C equals zero
you basically second equation equal to zero
with the terms in descending order of degrees
okay so this must be done first before you
determing what ABN CR why computer the discriminant
and also the solution of the quadratic equation
using the quadratic formula right so let's
go ahead and carry out how manipulation 3X
square +7 equals negative 2X is clearly not
in the standard form because of the can see
it's not equal to zero to transform it into
the standard form we simply add 2X to both
sides so this will be equal to zero in there
rearranged terms accordingly okay so we have
3X square plus 2X +7 equals zero so this ladies
and gentlemen is the standard form that's
we're talking about okay the beauty of the
standard form is that it enable us to determing
what a BNC are so you have this is your a
the coefficient of the X square be the coefficient
of X and see if you constant so let's go ahead
and write that out we have a equals 3B equals
two and see equals seven now let's find the
discriminant so is read the formula the discriminant
is b squared minus four A C (b^2-4ac) the
radical components of the Drabek formula the
ready can by the car out the substitution
you have two square -4×3×7 now one good
practice when you finding the discriminant
or using the quadratic equation is to use
parenthesis now the use of parenthesis it
extremely importance went in the of these
terms are negative okay you thoroughly see
the benefit here because the all positive
the whenever you have a negative term you
must use parenthesis in order to avoid making
sign errors or confusing yourselves in the
set up of the expression isolate go ahead
and simplify this using the order of operations
we square that get 4-4 times threes 2121×7
real quick correction four times threes not
21 this 3×74 times threes 12 so 57 okay right
so we have 4-12×7 is 8484-4-84 is -80 okay
so your discriminant Scriven and is -80 so
what is this tell us about the nature of the
roots was is having negative  discriminant
let's go back to our notes we say that if
we have a negative discriminant that you have
two distinct imaginary roots okay so let's
go ahead and write that down since the discriminant
the discriminant is negative we have to be
met two distinct imaginary roots right as
the solution to the quadratic equation of
the solutions to the quadratic equation okay
so that's what we should expect to gets when
we find the actual roots now let's go ahead
and not complete the problem I actually finding
the roots using the the quadratic formula
okay so let's some do that part two is to
find the roots using the quadratic formula
find the roots okay now when you have a quadratic
equation that see 3X square plus 2X +7+ we
have in this case there are different ways
that you can sold it okay in this vertically
keys were told to use the quadratic formula
now there are the methods that can be used
you can use the factoring method you can use
complete can the square we can use of graphing
method if you equipped with a graphing calculator
or some kind of graphing software now delimitation
of the factoring method is if you have a prime
quadratic equation that's the factorable then
you are stuck okay but the nice in about the
quadratic formula is that it always works
whatever quadratic equation you throw its
way you will always be able to find the roots
of that quadratic okay the us the limitation
of the complete can the square are going them
is that it's longan involved a lot of steps
involved there are the quadratic formula is
actually the condensed version of the completing
the square algorithm so if you know how to
use the quadratic formula you are in good
shape okay so let's go ahead and write down
what the quadratic formula is as elites X
equals negative the plus a minus the square
roots of b squared minus four A C (b^2-4ac)
divided by two a now do you see the discriminant
here this is the discriminant it in the rowdy
can of the quadratic formula okay so that's
what comes from our right so very know what
ABN CR so let's go ahead and  sold it by
the nice in about  finding the discriminant
is that we are you know what goes on the need
the square roots okay so we do negative be
20 put a BNC back here for a is 3B is to see
is seven so X is negative be negative to plus
a minus the square roots of b squared minus
four A C (b^2-4ac) so b squared minus four
A C (b^2-4ac) is -80 where it in one that
is so can just put it here will have to the
work again and the whole thing divided by
two a a is three so we have to times 30 okay
now this becomes negative to plus or minus
roots -80/6 now let's go ahead and see if
we can simplify this rowdy can okay so we
have root 80 -80 actually dietary have a negative
of the rowdy can negative number that minus
comes out as and I remember to square root
of negative one is I now will proceed to the
compose 80 into  product of primes and then
whichever factor repeats will will extract
as the roots okay so let's factor this out
can take out to from 80 to goes and 8040 times
take up another 22 goes into 4020 times they
cover number 22 goes into 2010 times taken
another 22 goes in a 10 five times okay now
any factored out repeats itself we can take
the square root of the product of those who
factors and this end up with one of them so
let us see what repeats we have 2×2 what
is the square root of 2×2 or to square root
of four squared of 2×2 is used to so those
two can come out us to and then we have another
repeated product of factors here what is the
squared of 2×2 or the square root of four
is also two now take a look at five if there
by itself so five cannot come out of the radical
because it's by itself okay so the final product
is going to be to square root of 2×2 know
have 2×2 roots five okay let's go ahead and
write that down right so we have Om to these
two twos come out of the to times these two
twos Kamath is another to the negative remember
it comes out of them I and the rat five stays
behind okay we multiply these two twos any
out so we have four I roots five that's the
reduced form of -80 okay so that it take them
simply plug it in this position right here
alright so let's go ahead in simplify further
we have X equals -2+ or -4 I roots five the
entire expression divided by two a now want
anyone be careful with is you have to divide
the entire numerator by 28 not just the radical
part okay now let's go ahead and reduce only
reducing will look at all three integers okay
so looking at -2 and looking at four and six
and there were going to ask ourselves is there
a number that goes into all three there greatest
one factor out can divide all three by okay
is as though it's all are not been we don't
have something a goes into all three then
we're down so what is the greatest common
factor of 24 and six the GCF is to so that
means that this expression can be simplified
to achieve that's will simply divide every
single integer by the GCF which is to the
rights you like to divide for right to and
divide 6 x 2 okay down give us the simplified
form of our results so that I have negative
one plus or -2 I 40 by like to us to roots
five all over 6÷3 6÷2 is three okay so let's
separate our to answers we can see the two
roots that we have the first imaginary roots
he have is negative one -2 I roots 5÷3 that
the first root the second root that we have
is negative one +2 I roots 5÷3 so we just
splitting this answer using these two signs
Psalm our bases for separation okay so these
are our to imaginary roots and we see why
the or imaginary because we have that I components
in the results okay so this is basically how
to use the quadratic formula to find the roots
of the quadratic equation thanks so much for
taking the time to watch this presentation
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