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How To Solve Quadratic Equations By Factoring - Quick & Simple!

in this lesson we're going to talk about

solving quadratic equations by factoring

so let's start with this example x

squared - 49 is equal to zero you can

use the difference of perfect squares

technique for this one the square root

of x squared is X the square root of 49

is 7 so it's gonna be X plus 7 and X

minus 7 now you need to set each factor

equal to 0 at this point and then you

just find the value of x so we have X

plus 7 is equal to 0 and X minus 7 is

equal to 0 the reason why we can do that

is because if one of these terms is

equal to 0 then everything is 0 0 times

anything is 0 so X is equal to negative

7 and in the other equation if we add 7

to both sides we can see that X is equal

to positive 7 let's try another example

let's say if we have 3x squared minus 75

is equal to 0 what is the value of x 3

and 75 are not perfect squares so we

don't want to use the difference of

perfect squares taken again however we

can take out the GCF the greatest common

factor which is 3 3x squared divided by

3 is x squared negative 75 divided by 3

is negative 25 now we can use the

difference of perfect squares technique

to factor x squared minus 25 the square

root of x squared is X the square root

of 25 is 5 so it's going to be X plus 5

and X minus 5 so if least set X plus 5

equal to zero we can clearly see that X

will be equal to negative 5 and if we

set X and minus 5 equal to zero X is

equal to plus 5

and so that's it for that one now what

about this one was saying if we have 9x

squared minus 64 is equal to zero

well first we can use the difference of

perfect squares technique we can square

root nine and we can square root of 64

the square root of nine is three the

square root of x squared is X the square

root of 64 is 8 so it's going to be 3x

Plus 8 3x minus 8 so if we set 3x plus 8

equal to zero then we can see that 3x is

equal to negative 8 which means X is

equal to negative 8 over 3 now if we set

3x minus 8 equal to 0 and solve for x X

is going to be positive 8 over 3 using

the same steps now what if we have a

trinomial x squared minus 2x minus 15

and the leading coefficient is 1 how can

we factor this expression all you need

to do is find two numbers that multiply

it's a negative 15 but that add to

negative 2 numbers that multiply to 15

or 5 and 3 so we have positive 5 and

negative 3 or negative 5 and 3 5 plus

negative 3 adds up to positive 2 but

negative 5 plus 3 adds up to negative 2

so this is what we want to use it turns

out that to factor it it's simply going

to be X minus 5 plus X plus 3 so if we

set X minus 5 equal to 0 X will be equal

to 5 and if we set X plus 3 equal to 0 X

will be equal to negative 3 let's try

another one like that

let's say if we have x squared plus 3x

minus 28 so what two numbers multiply to

negative 28

but as you three

go ahead and try so if we divide 28 by

one we'll get negative 28 if we divide

negative 28 by 2 negative 14 3 doesn't

go into it if we divide it by 4 we'll

get a negative 7 4 and negative 7

differs by 3 if we add them it's a

negative 3 so we need to change the sign

so it's gonna be X minus 4 times X plus

7 which means that X is equal to

positive 4 and negative 7

here's another problem

so how can we factor this trinomial when

the leading coefficient is not 1 so 1 we

need to do in this problem we need to

multiply 8 and negative 15 8 times

negative 15 is negative 120 now what two

numbers multiply it to negative 120

but at attune if you're not sure I make

a list

let's start with 1 we have 1 in 120 see

you 163 and 44 + 35 + 24 6 and twenty

eight and 15 now 10 and 12 seem

promising 10 and negative 12 differ by

negative 2 but positive 12 and negative

10 adds up to positive 2 so what we're

going to do in this problem is we're

going to be placed 2x with 12 X and

negative 10 X and then factor by

grouping in the first two terms let's

take out the GCF which is going to be 4x

8x squared divided by 4x is 2x + 12 X /

4x mystery and the last two terms take

out the greatest common factor in this

case negative 5 negative 10x divided by

negative 5 it's 2x and negative 15

divided by negative 5 that's +3 now if

you get two common terms that means

you're on the right track you can write

it once in a parenthesis in the next

line now the stuff on the outside the 4x

and negative 5 that's going to go in the

second parenthesis so that's what we

have

now let's set 2 X plus 3 equal to 0 and

4x minus 5 equal to 0 so in the first

equation let's subtract 3 from both

sides so 2x is equal to negative 3 and

then let's divide by 2 so the first

answer X is equal to negative 3 over 2

now let's find the other answer so let's

add 5 to both sides so we can see that

4x is equal to 5 and then let's divide

both sides by 4 so X is equal to 5 over

4 and that's it for this problem now

let's get some of the answers to the

quadratic equations that we had in the

last lesson so for this particular

problem when we factor it we got a

solution of 5 and negative 3 and less

than 10 point 2

but now let's use the quadratic equation

to get those same answers so X is equal

to negative B plus or minus the square

root of b squared minus 4ac divided by

2a

that's the quadratic formula and you

need the quadratic equation in standard

form so we can see that a is equal to 1

B is the number in front of X B is

negative 2 and C is negative 15 so let's

replace B with negative 2 B squared or

negative 2 squared negative 2 times

negative 2 is 4a a is 1 and C is

negative 15 divided by 2 a or 2 times 1

which is 2 negative times negative 2 is

positive 2 and then we have 4 negative 4

times negative 15 that's positive 60 and

60 plus 4 is 64

now the square root of 64 is 8 so we

have 2 plus so minus 8 divided by 2 2

plus 8 is 10 10 divided by 2 is 5 that

gives us the first answer the next one

is 2 minus 8 divided by 2 2 minus 8 is

negative 6 negative 6 divided by 2 is

negative 3 which gives us the second

answer so you can solve a quadratic

equation by factoring or by using the

quadratic formula now let's try another

example 8x squared plus 2x minus 15 use

the quadratic equation to find the

values of X so we can see that a is

equal to 8

B is the number in front of X that's 2 C

is negative 15 so using the quadratic

formula x equals negative b plus or

minus the square root of b squared minus

4ac divided by 2a so B is 2 which means

B squared that's going to be positive 4

minus 4 times a a is 8 C is negative 15

divided by 2a or 2 times 8 which is 16

so this is negative 2 plus or minus

square root 4 now negative 4 times

negative 15 is positive 60 60 times 8

that's 480 so we have 4 plus 4 80

so this is negative 2 plus or minus the

square root of 480 for the square root

of 4 84 is 22 so now we have negative 2

plus or minus 22 over 16 so now what

we're going to do at this point is

separate that into two fractions but

let's just let's make some space first

so this is negative 2 plus 22 over 16 or

negative 2

- 22 over 16 negative 2 plus 22 that's

positive 20 + 20 over 16 both numbers

are divisible by 4 20 divided by 4 is 5

16 divided by 4 is 4 so the first answer

is 5 divided by 4 negative 2 minus 22 is

negative 24 24 and 16 are both divisible

by 8 negative 24 divided by 8 is

negative 3 16 divided by 8 is 2 and so

that's the other answer negative 3 over

2 so now you know how to use the

quadratic formula to solve quadratic

equations