a

Solving Quadratic Equations by Factoring - Basic Examples

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