Math115

Notes

The

Quadratic

Equation

This

is

a

review

of solving

a

quadratic equation,

either

by

factoring

or

by formula.

Factoring

a

trinomial

expression:

A

trinomial

expression

takes

the

form

ax2

*bx*c.

Finding x

values

where

this

expression

is

zero

forrrs

a

quaAratic

equation.

The

solutions

for

x can

be

found

easiln

if the

"*pr"rrlon

factors.

If

not,

the

quadratic formula

must

be

used.

Factoring

method:

Take

6x2

-4x-2=0.

Since

there

is

a common

factor

of 2

(each

term

can be

divided

{y Z'lin

each

term

on

the

left

side,

divide

both

sides

by 2

to

make things

simpler.

We

get

3x2

-2y*l=0.

Sin".

the

last

term

is negative,

the

factors

have

different

signs

in the

centers.

There

is only

one

way

to

get 3x2

from

the

pioduct

of

the

two

l$

terms

of

each

factor.

Namely, 3x

times

x.

Since

1

is the

trirrtt

ofthe

produCt of

the

last

terms

of each

factor,

the

last

terms

must both

be

i. W"

must

try

different

ro*bi*tions

to

see

which

one

gives

us the

correct

middle

tenn

of -2x.

This

is

know

as

..trial

&

errot''

method

which

will

eliminated

certain

possibilities.

Try:

(3x-l[x+l).

Themiddletermherewillbe

-xaddedto3xforaresult

of

2x,incorrectsigt.

Try:

(3x+1)(x-1).

The

middle

term

here

will

be

*x

added

to

-3x for

a

result

of

-2* correcl

Note:

Not

all are

this

easy.Some

take

more

trials

to

achieve

the correct

result.

Now

that

we

have

the

factors,

we

are

solving:

(3x+1)(x-1)=0. Just

set

each

factor

=O

and

solve

the

simpler

equations

for

the

two

solutions

for x.

That

is, 3x+1=0

and

x-1=0

for

the solutions

of

'F-tl3

and

x-1.

,

If

it

were

the

case

that the

quadratic equation

did

not

factor,

we

would

have

to

use the

quadratic

fonnula

Aftertying

all

of

the

possibiiities

without

success,

we

would

conclude

that

the

expression

does

noifurtor.

This

could

be

time

consuming.

There

is a

nice

short-cut

that can

check

bei'orehand

if

the

expression

factors.

This

will

save

us

much

time. Use

the

following

prccedure:

Compute

b2

-4acand

see

if

it

is 0 or

a

perfect square.

This

expression

is called

the

"discriminate"

of the

quadratic

exprcssion.

In

otr

example,

using

the simpler

equation,3x'

-2x-l=0,

the

discriminate

would

be

(-2),

-4(3X-l)

:4+12=16,

a

perfect

sqture,

so

it factors

as

we

have seen.

However,

if

otg

quadratic equation

were

6x2

-5x-2{,

the

discriminate

would be

73,

not

a

perfect

square.

So,

the

two

solutions

for x

must

be found

by

the

following

quadratic

formula.

they are,

x:!+JE,

r5-Jfr

12

t2

y=-b+Jfro,

2a

and

x=

-b-

2a

Note:

In

the

above

case,

the

calculator

will

be

very

helpful

for approximating

the x

values.