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Algebra - Understanding Quadratic Equations

today we're going to take the mystery

out of quadratic equations so the

lecture today says understanding

quadratic equation so what are quadratic

equations well what they are is a

relationship between two variables

typically Y and X and it typically will

look something like this y equals ax

squared plus BX plus C now that's the

standard form as we call it a B and C

are simple constants any sort of numbers

X is the independent variable and then Y

is the dependent variable so as an

example let's write an equation such as

y equals let's say x squared minus 6x

plus 5 okay in this case a would equal 1

that's the coefficient from the x

squared term B is equal to minus 6 and C

is equal to 5 all right now one of the

things we want to be able to do with one

of these quadratic equations is learn

how to graph it another one is learn how

to find the roots but before we get into

all that let's kind of get a general

feeling of what acuatic equation is it's

again a relationship between x and y so

we should be able to graph it on an XY

plane so there's our y axis there's our

x axis and typically a chaotic equation

will be graphed either it will look like

this or it will look like that and the

way you can tell if it's going to open

upward or open downward as we call it is

if the number in front the x squared

term is positive it'd look like this

that's a plus a and if the number if

found the x squared term is negative

then it looks like this so think of it

as a minus 8 all right that helps us a

little bit and I also realize that if

you graph these that this parabola as we

call them these graphs for correct

equations called parabolas and it could

potentially for it to graph it it could

end up looking like this or can end up

looking like this or can up looking like

that or maybe it looks like this and

notice that in some cases the parabola

will cross the x-axis like this

particular parabola will cross it in two

places

this particular parabola will cross it

in two places and these two parabolas do

not cross the x-axis at all the places

where one of these parabolas which is a

graphical representation of a quadratic

equation like this where the parabola

crosses the x axis those are called the

roots of the quadratic equation so we

call these roots and here again

these would be considered the roots of

the quadratic equation notice that these

two do not have roots so sometimes when

they ask you to solve a quadratic

equation they're basically asking you

find the roots of the parabola and of

course if the parabola does not cross

the x-axis like again this is here the x

axis this here is the y axis like in

this case or in this case then there's

no solution there are no roots to the

quadratic equation all right so to help

us figure out what it actually is and

how to graph these quadratic equations

let's go ahead and factor this this

happens to be a factorable equation so

we can write this as y is equal to the

product of two binomials we write the X

and the X since there's a negative here

and a positive there that means one must

be plus and one must be negative or no

in that case I'm sorry actually they

both must be negative maybe because the

only way you can get a positive number

there is either that you have two

positive numbers here are two negative

numbers and since this is negative you

have to have two negative numbers so now

we're looking for two numbers when they

you multiply them you get a 5 when you

add them you get a 6 so looks like a 5

and a 1 because if I multiply x times

the negative 1 and multiply negative 5

times a an X and add them together I get

negative 6x my middle term alright now

if we want to solve a quadratic equation

to find the roots realize that the roots

are the points on

on the x-axis that means that that

location the y-value is equal to zero so

any point on the x-axis my y-value is

zero which means if I'm going to find

the roots I'm going to set my y equal to

zero if I do that

this equation now becomes zero is equal

to X minus 5 times X minus 1 and if I

solve this for X then I will find the

points where the equation crosses the

x-axis all right now in fact two

quantities multiplied together and the

solution is zero that means either one

or the other must be 0 because the only

way you can multiply two things together

and get 0 is if either X minus 5 equals

0 or the X minus 1 equals 0 and of

course if x equals X minus 5 equals 0

then X must equal 5 or if X minus 1

equals 0 then X must equal 1 and those

are the locations in this particular

example where the parabola will cross

the x-axis so if I'm going to graph what

I have here my example if I'm going to

graph this example on my XY axis here I

know that one of the roots or one of the

points where the the graph will cross

the x axis x equals 5 1 2 3 4 5 so right

there and the other place where it

crosses the x axis will be at x equals 1

which is right here and those are

considered the two roots of my chorionic

equation all right now there's another

thing about a quadratic equation that's

very important it's called the axis of

symmetry if I find the midway point

between those two numbers so this is 1 2

3 4 5 and so the midway between 1 and 5

is the number 3 if I now draw a dashed

line vertical line through the point x

equals 3 on the x axis that is now

called the axis of symmetry

and again if you look at my examples

over here if you draw a line halfway

between here and halfway between right

there and right in the middle there

right in the middle there right in the

middle there and right in the middle

there notice those lines those dashed

lines are the exact line that divides

the parabola in two equal parts

therefore that's called the axis of

symmetry now notice that the number from

the x squared term is positive it's a

positive 1 that means the parabola opens

upward that means I'm going to have a

problem that looks kind of like this to

find a few more details if I now plug in

to my equation the value of x where the

axis of symmetry goes right through in

this case the number 3 so I'm going to

solve for my equation when x equals 3

that's again the point where the axis is

similarly goes right through let's see

what I get so I'm taking my equation and

instead of X I'm going to write a 3 so I

get a 3 squared minus 6 times 3 plus 5

so notice I took my original example

instead of an X I write at 3 so this xri

2 3 and if I work that out I get 9 minus

6 times 3 is 18 plus 5 and so that's a

plus 14 minus 18 that's equal to minus 4

so that means when x equals 3 my Y will

be negative 4 so 1 2 3 4 that's negative

4 right here so x equals 3 y equals

negative 3 that is the bottom or top

point of my parabola so if my parabola

ups upward the lowest point of my

parabola is this point right there if I

my parabola opens downward

then my lowest or in this case my

highest point right there will be that

point right there so in this case since

a is a positive number my parabola opens

upward so this will be my lowest point

on my parabola also known as the vertex

so the vertex of my parabola now there's

one more special point about the

parabola

sometimes the parabola will cross the y

axis like over here in this case it

doesn't look like it but if it goes on

if you keep going long enough again

eventually you will cross the y axis

right here this parabola crosses the y

axis over here and if the probability

goes on long enough eventually you can

see way down here somewhere

the parabola will so cross the y axis

how do we find that point well remember

anytime you cross the y axis that means

that the x value must equal 0

so to find that particular point we take

our initial equation again and plug in 0

for X to see where the equation or where

the parabola crosses the y axis so I'm

going to solve for y when x is equal to

0 again I take my equation right there

plug in a 0 for every X that I find so I

get 0 squared minus 6 times 0 plus 5 or

y when x equals 0 is equal to 5 so going

to my example here my graph when x

equals 0 my Y value should be 5 so it's

1 2 3 4 5 and so my problem across that

point as well now notice I have four

points I know my equation can be graphed

like a parabola here's my lowest point

or the vertex there's my two roots

there's the point where the parabola

crosses the y axis if I now carefully

connect all those dots with a free hand

like that I have now drawn a graphical

representation of my example right here

my quadratic equation and you can see

how that looks like a nice parabola so

that's what a parabola is that's what a

quadratic equation represents and if you

want to look at the points very

carefully again notice you have the

lowest point on your parabola called the

vertex you have the two points where the

parabola will cross the x axis those are

also known as the roots

so that's a route that's a route right

there

also notice that most rabbits will cross

the y-axis and when they do that's the

point you can find you can find that

point by plugging in 0 for X in your

equation and then of course the axis of

symmetry runs right midpoint between the

two roots or also goes right through the

vertex so there you get a pretty good

feel for what a parabola is and for what

a quadratic equation is alright now

we're going to show you some examples of

actual how to solve these quadratic

equations and how to graph them in a

little more systematic fashion