### Video Transcript

In this video, weโre gonna look at
some examples of quadratic equations and use factoring, or factorizing, to find
their roots. Weโll also talk about some of the
different notations that you can use to present your answers. Weโve covered factoring different
types of quadratics in more detail in other videos, so weโll just quickly recap the
process here.

But first, remember that a
quadratic expression has a squared term, a linear term, and a constant term. So thatโs a positive or negative
number times ๐ฅ squared, a positive or negative number times ๐ฅ, and a pogi-
positive or negative constant on the end. Also remember that the coefficient
of ๐ฅ or the constant term on the end, so the ๐ or the ๐ value here, could either
be zero or they could both be zero for that matter. But the ๐-value here canโt be
zero, or otherwise it wouldnโt be a quadratic.

So for example if ๐ was one, ๐
was negative five, and ๐ was six, weโd have a quadratic expression of ๐ฅ squared
minus five ๐ฅ plus six. And also remember it doesnโt have
to be ๐ฅ; it could be any letter. The roots of a quadratic are the
๐ฅ-values which generate your result of zero, when you substitute them into the
expression. So for example a question like:
Find the roots of the quadratic equation ๐ฆ equals ๐ฅ squared minus five ๐ฅ plus six
means solve ๐ฅ squared minus five ๐ฅ plus six equals zero. In other words, find the values of
๐ฅ which match that equation.

Now in this case, letโs not worry
for now how we found them but there are two possible answers. When ๐ฅ is equal to two, we put
that number in and we get an answer of zero, so that works. And when ๐ฅ is equal to three, we
put that number in for ๐ฅ and we get an answer of zero, so that works as well.

Now we can present our answer in
two ways. We can either present it as a list,
so ๐ฅ is equal to two or ๐ฅ is equal to three; or we can present it as a solution
set, so two and three in set notation. And with quadratics you are likely
to come across this idea of real numbers, so our solution set is from the set of
real numbers. So thinking back to what our
quadratic graphs look like, if the graph cuts the ๐ฅ-axis in two places, then weโve
got two solutions. If it touches the ๐ฅ-axis in one
place, then weโve got one solution. And if it doesnโt cut through the
๐ฅ-axis at all, then weโve got no real solutions. If we use imaginary or complex
numbers then we can come up with some solutions, but weโre not gonna delve into that
right now.

Okay. Letโs look at some examples. So number one: Find the solution
set of ๐ฅ squared plus ten ๐ฅ equals zero. This might also have said: Find the
roots of ๐ฆ equals ๐ฅ squared plus ten ๐ฅ.

Now this is an equation which
easily factors or factorizes. Weโve got ๐ฅ and ๐ฅ in here, so ๐ฅ
will be our common term that we can take out. So we can express this as ๐ฅ lot-
times ๐ฅ plus ten is equal to zero. Now that ๐ฅ right up against the
bracket there means itโs ๐ฅ times ๐ฅ plus ten, so weโve got something times
something is equal to zero. Now the only way you can get an
answer of zero when you multiply two things together, is if one of those is
zero. So either ๐ฅ is equal to zero, or
๐ฅ plus ten is equal to zero. And if I subtract ten from each
side of that equation, Iโm left with ๐ฅ is equal to negative ten. So if I was just writing those as a
list, I would say my answer is ๐ฅ equals zero or ๐ฅ equals negative ten. But the question has asked for a
solution set. So using set notation, the two
values that ๐ฅ can take are zero and negative ten; so this would be my answer.

Number two: Find the solution set
of ๐ฅ plus seven all squared is equal to zero.

Now ๐ฅ plus seven all squared means
๐ฅ plus seven times ๐ฅ plus seven. So weโve got something times
something is equal to zero. Again something times something
equals zero, the only way we can get that is if one of those things is equal to zero
itself. So either ๐ฅ plus seven is equal to
zero, which would make ๐ฅ equal to negative seven, or the same thing. So weโve got repeated roots. This is one of those situations
where if we drew the graph of this function ๐ฆ equals ๐ฅ plus seven all squared, it
would look roughly like that. It would touch the ๐ฅ-axis in one
place at negative seven, and in fact it would cut the ๐ฆ-axis at forty-nine. So our solution set has only got
one item in it, negative seven.

Moving on to number three then. Find the solution set of ๐ฅ squared
plus two ๐ฅ minus thirty-five equals zero in โ, the set of real numbers.

So itโs just worth remembering that
just ๐ฅ squared means one ๐ฅ squared. And when weโve got one ๐ฅ squared,
thereโs a certain set of rules that we can follow when weโre doing our
factoring. So remember if weโve got ๐ฅ plus a
number times ๐ฅ plus another number and we multiply them out term by term, weโve got
๐ฅ times ๐ฅ which gives us ๐ฅ squared, and weโve got ๐ฅ times ๐ so thatโs positive
๐๐ฅ, and ๐ times ๐ฅ which is positive ๐๐ฅ, and ๐ times ๐ which is positive
๐๐.

Now looking at this weโve got ๐๐ฅ
and ๐๐ฅ, so thatโs factoring out the ๐ฅ there; weโve got ๐ plus ๐๐ฅ. So in converting between these two
different forms of our expression, weโve got, on the end weโve got ๐ times ๐ so
this number times this number. And here, as the multiple of ๐ฅ,
weโve got ๐ plus ๐. So Iโm going to add ๐ to ๐. So that should help us to do our
factoring remembering that-that format.

So Iโm looking for ๐ฅ plus or minus
something times ๐ฅ plus or minus something else. And if I multiply those two numbers
together, Iโm gonna get negative thirty-five and if I add them together, Iโm gonna
get positive two. So first of all, Iโm just gonna
write out all the factors of thirty-five.

So always start up with one and the
number, so one times thirty-five, two isnโt a factor, three isnโt, four isnโt, five
is a factor, five times seven, six isnโt a factor, and now weโre up to seven. Weโve already encountered seven in
our list, so we know weโve got all of the factors that we need; one times
thirty-five or five times seven. Well theyโre factors of
thirty-five. Now to get negative thirty-five,
one needs to be positive and the other needs to be negative. So just bare that in mind, one of
these is gonna be positive, the otherโs gonna be negative. Now when I add those two factors
together, theyโve got to give me positive two. Now for one and thirty-five, the
difference is thirty-four. So it doesnโt matter which one of
those is positive and which one is negative. When I add them together, Iโm never
gonna get an answer of positive two. But with five and seven, their
difference is two. So Iโve gotta think carefully which
one has to be positive, which one has to be negative.

And when I add them together, the
result is positive. So the bigger number needs to be
positive and the smaller number will need to be negative. So ๐ฅ minus five times ๐ฅ plus
seven is equal to zero. Now letโs just check that we get
the same expression when we multiply those out. ๐ฅ times ๐ฅ is ๐ฅ squared, ๐ฅ times
seven is seven ๐ฅ, minus five times ๐ฅ is minus five ๐ฅ, and minus five times
positive seven is minus thirty-five. And simplifying this down, seven ๐ฅ
take away five ๐ฅ is two ๐ฅ. That leaves us with ๐ฅ squared plus
two ๐ฅ minus thirty-five. Thatโs what we were looking for
originally so we-we have factored it correctly.

So we find ourselves in the
situation where weโve got something times something is equal to zero. And if something times something
equals zero, one of those things must be zero. So either ๐ฅ minus five is zero,
which would mean that ๐ฅ would be five; or ๐ฅ plus seven would be zero, in which
case ๐ฅ will be equal to negative seven. So our solution set consists of
negative seven and five.

Number four: Find the solution set
of four ๐ฅ squared minus twenty-five equals zero in the set of real numbers.

So hereโs a quadratic which has a
๐ value of zero. And in fact, this is a very special
case because four ๐ฅ squared can be written as two ๐ฅ all squared, and twenty-five
can be written as five squared. So weโve got the difference of two
๐ฅ- of two squareds; two ๐ฅ all squared minus five squared. Now letโs just take a second to
remember the difference of two squareds technique. If I had two parentheses here ๐ฅ
minus ๐ times ๐ฅ plus ๐, ๐ฅ times ๐ฅ is ๐ฅ squared, ๐ฅ times positive ๐ is
positive ๐๐ฅ, negative ๐ times ๐ฅ is negative ๐๐ฅ, and negative ๐ times positive
๐ is negative ๐ squared. So multiplying out those
parentheses, Iโve got ๐ฅ squared plus ๐๐ฅ take away ๐๐ฅ. So these two terms, if I start off
with ๐๐ฅ and I take away ๐๐ฅ, thatโs gonna give me zero; so thatโll cancel
out. And then Iโve got minus ๐
squared. So Iโve got ๐ฅ squared take away ๐
squared; the difference of two squareds. So Iโm gonna use this result ๐ฅ
minus ๐ times ๐ฅ plus ๐ gives me ๐ฅ squared minus ๐ squared, in order to help me
factor this expression here.

So something squared minus
something else squared gives me the something minus the something else times the
something plus the something else. So Iโm gonna do the same pattern
here. So this is what we get factored and
remember from our original equation that that is equal to zero. So weโve got something times
something is equal to zero. So either two ๐ฅ minus five is
equal to zero, which when I rearrange and solve that equation I get ๐ฅ is equal to
five over two; or two ๐ฅ plus five equals zero, which I can rearrange and solve to
get ๐ฅ is equal to negative five over two. So here are my two answers: ๐ฅ is
five over two or negative five over two. And using set notation like it
asked for in the question, my solution set of real numbers is negative five over two
or- and five over two.

Lastly then number five. Find the solution set of six ๐ฅ
squared plus eleven ๐ฅ minus ten equals zero in the set of real numbers.

So here weโve got to factorize a
quadratic expression which doesnโt have one ๐ฅ squared; itโs got more than one ๐ฅ
squared there. So we need to factor that
quadratic, put it equal to zero, and see what two solutions we get. Now this is a bit trickier than
those monics. So the word monic means when the
coefficient of the highest power term is equal to one. So for example one ๐ฅ squared plus
two ๐ฅ plus five. Now as we said, these monics were
easier to factorize than these non monics. So letโs just go through a method
then of factoring these non monic quadratics.

So first of all Iโm gonna do the
coefficient of ๐ฅ squared, six, times the constant term negative ten. And six times negative ten is
negative sixty. Then what Iโm gonna do is write out
all the factors of โ well Iโm just gonna do sixty and then weโll deal with the minus
sign later on. So one times sixty is sixty, two
times thirty is sixty, three times twenty, four times fifteen, five times twelve,
six times ten, seven isnโt a factor, eight isnโt a factor, neither is nine, and ten,
weโve already encountered ten; so weโve got all of our factors of sixty.

So Iโve got six pairs of factors of
sixty. But what I need to know now is
which pair of those, when I add them up, do I get positive eleven; the coefficient
of the ๐ฅ term. Now remember because theyโre gonna
multiply together to make negative sixty, one of them is gotta be positive, one of
them is gotta be negative. So when I add them together, Iโm
gonna get the difference of the two. So Iโm looking for two factors that
have a difference of eleven. So obviously one and sixty isnโt
gonna have a difference of eleven, two and thirty isnโt gonna have a difference of
eleven, nor is three and twenty, four and ~~five~~ [fifteen], they do have a
difference of eleven. So I need to work out which one
needs to be positive and which one needs to be negative. Iโm trying to generate positive
eleven, so the biggest one of these has to be positive and the smallest one needs to
be negative. So negative four plus fifteen gives
me neg- gives me positive eleven.

So Iโm now gonna re-express this
middle term plus eleven ๐ฅ as a combination of negative four ๐ฅ and positive fifteen
๐ฅ. Now it doesnโt matter which way
around I write that, positive fifteen ๐ฅ take away four ๐ฅ, or negative four ๐ฅ add
fifteen ๐ฅ; so Iโve chosen to do it this way. But basically, fifteen ๐ฅ take away
four ๐ฅ gives me eleven ๐ฅ like we got in the line above; so those two lines are
completely equivalent. So having re-expressed that middle
term, the ๐ฅ, Iโm now gonna treat this as two separate halves, six ๐ฅ squared plus
fifteen ๐ฅ and negative four ๐ฅ take away ten ๐ฅ. And Iโm gonna factor the first
half. So six and fifteen have got a
highest common factor of three, and ๐ฅ squared and ๐ฅ have got a highest common
factor of ๐ฅ. So thatโs gonna be three ๐ฅ times
two ๐ฅ. Cause three ๐ฅ times two ๐ฅ gives
me six ๐ฅ squared, what do I need to multiply three ๐ฅ by to get fifteen ๐ฅ; thatโs
just five.

So Iโve just factored the first
half of that expression up here. Now this thing in brackets, two ๐ฅ
plus five, I want this to be a common factor. So Iโm just gonna write that
bracket out again here, two ๐ฅ plus five. And I need to work out what do I
need to multiply that by in order to get this expression up here, negative four ๐ฅ
take away ten. So two ๐ฅ, what do I need to
multiply two ๐ฅ by to get negative four ๐ฅ. Well that would need to be negative
two, negative two lots of two ๐ฅ is negative four ๐ฅ. So letโs just check now that
negative two times positive five, that gives us negative ten. Yep, thatโs the other term that
weโre looking for.

So these two lines are also
equivalent, weโve just rewritten it in a slightly different way. Instead of six ๐ฅ squared plus
fifteen ๐ฅ, weโve got three ๐ฅ lots of two ๐ฅ plus five and instead of negative four
๐ฅ take away ten ๐ฅ, weโve got negative two times two ๐ฅ plus five. Now Iโve got something times two ๐ฅ
plus five take away something times two ๐ฅ plus five. That two ๐ฅ plus five is a common
factor to these two terms. So Iโm gonna take that out as ano-
as a factor. Now in the first term, I had two ๐ฅ
plus five times three ๐ฅ. And in the second term, I had two
๐ฅ plus five times negative two. So I factored my quadratic
expression down to two ๐ฅ plus five times three ๐ฅ minus two. And if we do a quick check of that
two ๐ฅ times three ๐ฅ is six ๐ฅ squared, two ๐ฅ times negative two is negative four
๐ฅ, positive five times three ๐ฅ is fifteen ๐ฅ, and positive five times negative two
is negative ten. And as we said before negative four
๐ฅ plus fifteen ๐ฅ is plus eleven ๐ฅ. So yep, that is the expression we
were looking for. So it looks like we did factorize
it correctly.

So doing our factoring, weโve got
something times something is equal to zero. So one of those is gonna be
zero. So either two ๐ฅ plus five is zero,
in other words ๐ฅ is equal to negative five over two; or three ๐ฅ minus two is zero,
in other words ๐ฅ is equal to two thirds. So our solution set is negative
five over two, or two thirds. And remember if we were writing
this as a list, we wouldโve written ๐ฅ is negative five over two, or ๐ฅ is equal to
two thirds.

So to summarize then, roots are
๐ฅ-values that substitute to give the expression a value of zero. Quadratics can have two, or one, or
no real roots. Then you need to choose a method of
factoring. Whether youโve got simple
factoring, difference of two squareds, a monic, or a non-monic, you can factor into
two brackets. And when you put those equal to
zero and youโve got something times something equals zero, either the first thing is
equal to zero, or the second thing is equal to zero. So that enables you to find your
solutions. Enjoy your factoring to solve your
quadratics.