Solution of a Quadratic Equation by Factorization Video Lecture - Class 10

all right in this video I want to do a
couple examples of solving some
quadratic equations by factoring and in
our first example here we've got x
squared minus 9x plus 14 equals 0 and
again the basic idea is we're trying to
factor first off so one thing that's
important is you want your one side of
the equation to be equal to zero so
notice we have that in our first example
we don't have that in our second example
so that's actually something that we'll
have to do first in that problem but in
this one it is set equal to zero so to
me that's that's good and again since
the coefficient on the x squared is a 1
if it factors at least you know nicely
with whole numbers I think to get the x
squared I would need an X and an X and
then again I'm looking for two numbers
that multiply to positive 14 but add up
to negative 9 so you know I'm thinking
about factors of 14 in my head so you
know 1 in 14 well I don't see how I can
you know add or subtract or do anything
with that to get a negative 9 so I don't
think that pair is gonna work well 2 & 7
I can multiply those together to get
positive 14 so and hey you know 2 & 7 if
I add those right now I would get
positive 9 but if we make them both
negative so negative 2 and negative 7
those still multiply to positive 14 but
now they'll give us the correct sign in
the middle so the way I think about it
if the if the whole number if the number
is positive they either have to be both
positive or both negative whatever the
sign on the middle is if it's negative
that's what they'll both have to be okay
so we use negative 2 and negative 7
equals zero okay so the point of having
this factored
is now you know again you're trying to
we want to figure out some value for X
so if we plug it into the first set and
then we plug it into the second set when
we multiply those two resulting numbers
we want to get zero out but if you
multiply two numbers and get zero out it
has to be true that at least one of
those two numbers is zero not that X is
zero but at the end of the day when you
do the arithmetic you have to get zero
inside the parentheses so what that
tells me to do is it says well really we
either want to make X minus two equal to
zero we want that to work out and be
zero or we'd have to take what's in the
second set of parentheses X minus seven
and set that equal to zero so this is
the idea you factor you take you know
each set of parentheses set it equal to
zero and then try to solve that equation
so in both in the first case we can just
add two to both sides we'll get x equals
positive two we can add 7 to both sides
on the second one and get x equals seven
and now we have our two solutions to
this quadratic equation again a
quadratic equation can have exactly two
solutions or it can have exactly one
solution or it can have no solutions at
all I think both of my examples have two
solutions
so I write 2x squared minus 5x equals 3
so again the first thing we're gonna
have to do is make one side equal to
zero a big common mistake so this is
something you don't you know don't want
to do
a common mistake is people will say oh
you know well they'll just go ahead and
start factoring and they'll say hey
there's an X you know present in both of
these terms so let's factor the X out
and then we have I guess it would be 2x
minus 5 if you distribute that you'll
get 2x squared minus 5x equals 3 and
then they'll take each little part and
say well either X has to equal 3 or 2x
minus 5 has to equal 3 and then they'll
solve this for X they'll add 5 and get
2x equals 8 and divide by 2 and get x
equals 4 and that is very bad because
what you're saying now is you're saying
if you have some number multiplied by
another number by making this statement
you're saying well either the first set
of parentheses has to be the number 3 or
at the end of the doing the arithmetic
what's in the second set of parentheses
has to equal 3 and again that's just not
true
you can certainly multiply two numbers
together or neither of them is 3 but at
the end you do get 3 is the answer so
they just if you think about it you know
this at all doesn't make any sense to do
okay again when it's equal to zero to me
it makes perfect sense to be able to
break them down and do these two
individual equations so I think that's
something kind of important to think
about because again then you're not
memorizing some arbitrary procedure
you're kind of you're understanding it
at that point why why it is what it is
okay so in the first example again we
want to make one side equal to zero so
I'm simply going to subtract 3 from both
sides we'll get 2x squared minus 5x
minus 3 equals 0 and now to me this is a
little tedious because this coefficient
of a 2 in front of the x squared
we have to be more careful when we
factor I think this is one I could
almost do by trial and error though you
could always do factoring by grouping
but I'm going to just try to trial and
air it so to me the only way to get the
2x squared at least using whole numbers
I definitely need an X and an X to get x
squared I'm gonna have to have a 2 and a
1 to get the two okay fair enough to get
negative 3 I'm either gonna have to use
positive 1 and negative 3 or positive 3
and negative 1 again if I'm using whole
numbers and you know in my head I just
start kind of playing with them so you
know because again I have a choice also
where they go but I think if we put the
minus 3 out here and the plus 1 inside I
think this is gonna work because we'll
get our 2x squared notice 2x and
negative 3 will give us our negative 6x
and then we'll get plus 1x which will
give us negative 5x and then we still
get our negative 3 so again you know try
factoring by grouping if you don't like
this method certainly there's other ways
to solve these equations as well but
again we're talking about factoring here
so same idea I take each little set of
parentheses set it equal to 0 and then I
just solve those resulting linear
equations and in these cases so I would
subtract 1 from both sides I might on my
first equation so I would get 2x equals
negative 1 and then I would divide both
sides by 2 and I would get x equals
negative 1/2 is one of my solutions for
the other one I would simply add 3 and
get x equals positive 3
and I've got my two solutions to that
original quadratic equation