# Algebra 1 – Lesson 9.4 Factoring to Solve Quadratic Equations

or unless nine point four factoring to
solve quadratic functions this kind of accommodation of what you did in chapter eight where you
factored polynomials and what you’ve done in chapter nine
brief find the solutions to quadratic functions the learning target today’s to solve
quadratic functions by factoring so set up your cornell notes and use this as your central question just a quick review from yesterday where we learned that these are the roots and remember these routes are where
white equal zero so really if you wanted to find the
roots of a quadratic equation like this one secret blah why equals zero and all your to do is figure out what
value of tax makes this equation equal to zero today what you’re going to do is take a
look at some quadratic equations but you first have to remember a rule that’s known as the zero products
property role and basically what it means is if you’re
multiplying two members together and uh… two numbers modified to gerry
cool zero what’s true is either this equals zero or this equals zero or the both could be
called as europe but it only takes one zero animal gation problem to make the whole
thing eight equal to zero so five seven time zero people zero five zero times for equal zero so just keep in mind and here we go the first problem says what are the
solutions of the equation forty plus one t_-minus to equal zero so what you’re trying to figure out is
what is the value of tea that makes this equation equal to zero and what’s true is here four t plus one past equal zero or t minus to past equals zero all you have a car to simple equations
that you can solve forty plus one equals zero t_-minus to equal zero a minister with the easier of the to the
t_ minus two equals zero to solve this yet to to both sides t equals two the first one it’s a little more
competative but not really subtract one from both sides forty who’s made of one divided by four t equals negative one force so the solutions to this decreasing disquiet recuperation people’s negative one fourteen antique was too and that’s it i don’t know what the problem like if i
were to craft this but i do know they would cross at too and negative one force so it might look something like that so take a couple seconds here’s some more practice problems hit the pause button and when you’re
finished hit play are showing the answers okay to solve these all i’m gonna do it
said each of these binomial ps equal to zero explodes one equals url annex minus five equals zero why ’cause in order for this product to
these two things be most blood together in order for this to be equal to zero either ex post one equal zero or xmas
five equals zero so i subtract one on this first equating
agate exit was negative one and i had five and they x because five so the solutions
to this problem are negative one and five in the second from part b i do the same thing i’d just take each
of these binomial since it equal to zero in the first one i subtracted three from
both sides i get two xy cause negative three and then divide by two in the second one i had four to both
sides and that’s my answer x equals four so the solution for this one negative three outs and four and you can see sisters simple solving of algebraic equations about this tibet easy that would be
great we’d be done but the next equations shes a little bit
more difficult what you have is a quadratic equation in the form xc where’d plus eight ex
post fifteen equals zero my question is one of the solutions so this system of the few lecture steps expert posted expose fifteen equal zero
i’m gonna start by factoring that i’m in a factory just like i did in
chapter eight commence at my tech tech job board two numbers a multi to give me x where x
times acts two members of both by to give me
fifteen are five times three and when i crossed multiply i get five x plus reacts with cheap eagles a
tax so this part of this project function in factored form looks like this expose five annex plus three and that’s equal to zero one other matter in in fact reform i can
tell you one of these past equal zero to make the whole thing
zero so i said each one equal to zero that will give me the solutions subtract
five from both sides again x equals figure five and the second when i said recovery from
both sides and i get back sequels may give you three the solutions to this equation x squared
plus aid exposed fifteen equal zero are exit was negative thought of annex he was a good three those of the roots this is a quadratic equation and when you graphic widrick equation it
makes a probable and by doing what i’ve just done i can
tell you that the probable process negative five and it negative three so if i were a graph it one to three there’s the first route for five there’s the second it will look something like this now we’re gonna get a little more
specific in future lessons and graphic completely but this just
give you a picture of what it looks like again here are a couple more problems
for you to work out on your own remember you can start by factoring and then saying to me each equal to zero so hit the pause button and when you
finished culture if that’s what should cake in part a first thing to do is to factor in square
minus five m man’s fourteen so you get am times m is m squared native seven times positive to is
negative fourteen and when you cross both by you get
negative seven a_m_ plus two m his negative five m so this factors into am minus seven and am plus too when you set each of these binomial ps
at minus seven and employs two equal to zero and yourself at seven to both sides picket ami kal
seven subtract two from both sides you get am
equals negative too ami kal seven and then because they get
to writers solutions for this quadratic function in the second one p squared plus p minus twenty pete m spy’s p squared negative five times positive for his
native twenty hoops to get that rob positive five
times in a euphoric was may of twenty and when you cross multiplied at five p and a two to four p added together makes positive one p so if you write this out in factored
form egypt p plus five times p minus four equals zero set each of these binomial sequel to zero solve each equation peak was negative five and p equals positive for and once again what these solutions give
u are the roots or where the problem crosses the x-axis so if you drew a
picture of this one here’s native to one two three four five six seven here’s
positive seven you could look something like that little more detail involved but it gives you general picture one two three four five physicist
negative five one-two-three-four aniket looks something like that mess problem too in the third problem he pretty much
doing the same thing we did in problem too there’s just one little extra piece and
that is in the first problems that we’ve done he’ll notice it’s already set equal to
zero in this example you’re given in equation but it’s not
equal to zero if you just have to make it equal to zero by subtracting eighteen from both sides so you get forex squared minus twenty one x minus eighteen equals zero and now you’re pretty much doing the
same type of problem they didn’t problem too you’re gonna factor you’re gonna get too binomial ps equal to zero and then you’re gonna finish the problem so that the pos one and give this one a
trial by yourself and i’ll give you a solution in just a second toh kehta factor for exquisite ministry
one x minus eighteen animals by four x time sixty four x
squared animals by times native six to get negative
eighteen when you cross multiplied three x and four times in eighty six is made of
twenty four x add those together you get me a twenty
one x of this polynomial in fact reform is forex plus three and x minus six and figure out the solutions either one of these po binomial speedy equal zero serious ethical to zero and then yourself the first one subtract three from both sides you get forex minus three finish it by dividing by four annie get
x equals negative three-fourths there’s the for solution sector ones much easier had six to both sides you get x equals
six and those of the routes or solutions of
uh… he cleaned for experiments where one xy
causing team well asked problem is probably the most
difficult most because it’s a story problem it also involves a little more work but here we go you’re constructing a frame for the
rectangular photo shown you want the frame to be the same with all the
way around and the total area of the frame and
photo to be three hundred fifteen square inches what should the outer dimension of the
frame be all in other words when you look at this
picture i wanna know what access to be here which is that dimension or the width of
the frame all the way around so if you look closely might be
difficult to see on your screens already in in a different color the height of the picture has eleven inches and the with and seventeen inches that means told a mention of the whole frame and picture from left to right his acts plus seventeen prospects or to x pa seventeen and that means total height of the
picture is packs plus eleven plus x or to expose eleven that area of this rectangular photo with
the frame is to expose eleven times to expose seventeen and that equals three hundred fifteen so in this problem really we have to do
is take these two bynoe mills multiply them
together ghetto all the pieces together and say
legal two zero and then go backwards so it’s kind of problem where you go
forward and then backward to get the answer here’s what i mean you’re gonna multiplier out each of these binomial ps and you get to extends to exes for x squared to extend seventeen it’s thirty four x eleven times two x’s twenty two x and eleven ten seventeen his one eighty seven and that’s because of the three hundred
fifteen so i mean adn like terms thirty four ext
plus twenty two x it’s fifty six packs hoops not equals i’m still adding to that four x squared i’m still having one hundred eighty
seven and that’s still equal to three fifteen now the other problems when we had this
type of equation we had to set it equal to zero to find the roots so i’m gonna
take at three fifteen and subtracted from both sides to get equal to zero and when i end up with is four x cleared plus fifty six x minus one hundred twenty eight and now you have a quadratic equation equal to zero that you can factor so that your end up with two binomial xs set equal to zero and then you can t
each piece and said it equal to zero same steps as before but like i said it’s just a little bit
longer way to go through all of the smoky shin at a like terms subtraction three
fifteen from both sides what’s greatness factor this sequence and one thing you should notice or could
notice right off the bat is thirty six and one twenty eight share a
common factor of four simona factor that out first to
make my other factoring easier times xc whereas forex where’d plus four times fourteen x is fifty six x minus thirty-two now factory next witness fourteen x
minus thirty two is way easier than the first equation ticket that i mocked by x times x which
is x where’d to get naked thirty-two animal to by sixteen times negative to which is near thirty
two cross multiply i get sixteen x and negative two x which is positive fourteen x which is
what i want that means my factors are explicit steen annex minus two i can put the four out here cody cracked but it doesn’t change my
final answer if i set explicit teeny co zero annex minus chillicothe zero i’m gonna get an answer of negative
sixteen and positive q now that’s great if all you have is an
equation but this isn’t just an equation this is a story problem let’s go back up and remind ourselves
what it is we’re working on the question is what should outer dimension of the frame be we have two answers which one is the
best answer well you can’t really haven’t outer frame
that’s negative sixteen inches so you have to go to positive answer
which is to so let’s make sure we answer the
question correctly it’s inches switch it down to mention b so the correct answer is powder to mention should be inches and that’s it soul really quick review when you’re finding solutions to a
quadratic equation minutes already in factored form like this one is just said each binomial equal to zero
and solve the equation if it’s not in fact reform manure for
step is to take the quadrant equation factor it get an answer with two binomial set to
meet with a zero and get the solutions sometimes you’ll get an equation that’s
not set equal to zero so your first that has to be to get a kick with his erlich
we did in this example and then finally you can use the same
principles in-store problems like this one to figure out how big your frame should be okay uh… some of her lesson nine point
four is shown on the screen make sure you make any marks on your
notes to remind you to ask questions when you get back to class and that’s it