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- pooja195

- pooja195

Chapter 9 ;-;

- mathmate

Chapter 9
======

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- mathmate

|dw:1433627905280:dw|
Find x (hypotenuse) shown in the above drawing.

- pooja195

a^2 + b^2 = c^2?

- mathmate

Yep, now evaluate x numerically, please.

- pooja195

12^2+5^2=c^
144+25=169
13^2

- mathmate

So x=13 (is the final answer).

- mathmate

which two consecutive integers does sqrt(200) fall between?

- pooja195

idk this ;-;

- mathmate

10^2=100
11^2=121
12^2=144
13^2=169
14^2=196
15^2=225
16^2=256
...
which two consecutive integers does sqrt(200) fall between?

- pooja195

no .-.

- pooja195

*none

- mathmate

"between" is the keyword.
We know that 14^2=196, and 15^2=225
or sqrt(196)=14, sqrt(225)=15, so sqrt(200) falls between which integer numbers?

- pooja195

Decimals? .-.

- pooja195

wait no

- mathmate

no, we are looking for two integer numbers! lol

- pooja195

This si confusing T_T lets skip this

- mathmate

the answer is 14 and 15.
In fact sqrt(200)=14.1421356237309.... (never ends).
So we know that sqrt(200) falls between 14 and 15!
You were probably thinking that it's more complicated than this!

- pooja195

-_-

- mathmate

The hint is 14^2=196, so too small,
15^2=225, so too big.
Therefore sqrt(200) must fall between 14 and 15. Is that ok?

- pooja195

yes

- mathmate

Solve the equation 27-3y^2=0

- pooja195

Would you like the work or is it ok to put in the answer?

- mathmate

put in the work, but LaTeX is not required.

- pooja195

first find a GCF
27-3y^2=0
-3 is the gcf
then square root
answer: −3(y+3)(y−3)
set values to 0
y=3 or y=−3

- pooja195

/.\

- mathmate

Very good, shows that you're comfortable with factoring.
It will help in the later sections.
You can also solve by isolating y,
-3y^2=-27
y^2=9
y=\(\pm 3\)

- mathmate

Ready for the next one?

- mathmate

an engineering student is a contestant in an egg dropping contest. The goal is to create a container for an egg so it can be dropped from a height of 32 feet without breaking. The model for the egg's height, h (in feet), at time t seconds since release, is
h = -16t^2 + 32.
Calculate the time at whic the egg is at a height of 10 feet above ground. Give answer to 2 decimal places.

- pooja195

10=-16t^2+32
subtract 32 from both sides then divide by 16
1.375= t^2
idk where to go after this

- mathmate

Take out your ti-83 to finish!

- mathmate

"give answer to 2 decimal places" is a hint you might need your ti-83!

- mathmate

sqrt(1.375)=?

- pooja195

1.172604

- mathmate

exactly! (1.17 for 2 decimal places).

- mathmate

Any questions before we move to simplifying radicals?

- pooja195

no

- mathmate

Simplifying radicals basically is to pull things out of the square-root sign, whenever possible.
For example, sqrt(4)=2 is rather straight-forward.

- mathmate

But sqrt(18) is less obvious, since we write
sqrt(18)=sqrt(3^2 *2)=3sqrt(2)

- mathmate

So I'll let your try sqrt(48) while I take a meal break!

- pooja195

48
6 8
3 2 42
\[2\sqrt{12}\]

- mathmate

almost, just one step further
sqrt(48)=sqrt(4*12)=sqrt(16*3)=sqrt(4^2 * 3)=4sqrt(3)

- mathmate

Simplify
(1) sqrt(98) [98=2*7*7]

- mathmate

So every factor that appears twice inside the radical you take one outside.
So sqrt(98)=sqrt(2*7*7)=7sqrt(2)

- pooja195

Dance club.

- pooja195

Dance club is the method my teacher taught me because teens like to think like that

- mathmate

Oh! Can you show me that?

- pooja195

Ok its been a while but eh :P
56
8 7
4 2
2 2
2 and 2 are a couble so they leave the dance club
2 squrt 2
idk if its right

- mathmate

Yes, I get the part for the leaving dance club part (ingeneous!)
Then we have to keep 2*7 in the club, so we get
2sqrt(14)

- pooja195

lel :3

- pooja195

Are we done with this section ?

- mathmate

I usually like to see you do at least one perfect answer, like
(2) sqrt(60) [60=2*2 * 3 * 5]
then there is rationalize the denominator.

- pooja195

T_T

- pooja195

ok

- mathmate

(2) sqrt(60) [60=2*2 * 3 * 5]

- pooja195

im confused whats the question? .-.

- mathmate

simplify sqrt(60)

- pooja195

60
10 6
5 2 3 2
2 sqrt 15

- pooja195

T_T say im right!

- pooja195

Princess demands it!

- mathmate

Yes, you're right, I don't think you have problem with that.

- mathmate

Now we move onto rationalizing the denominator.

- mathmate

Mathematicians traditionally do not like to see square roots in the denominator, because that would make the common denominator very messy.

- pooja195

Just multiply by the root

- pooja195

*in the denominator

- mathmate

Do you want me to explain, or jump to pretest?

- pooja195

Pretest

- mathmate

Rationalize 9sqrt(1/3)

- pooja195

9 squrt 3

- pooja195

^LOL

- mathmate

Sure?

- mathmate

The question was \(9\sqrt{\frac{1}{3}}\)

- pooja195

:/

- pooja195

this is diffrent .-.

- mathmate

Example:
\(sqrt(\frac{2}{5}) = \frac{\sqrt2}{\sqrt5}=\frac{\sqrt2\times \sqrt5}{\sqrt5^2}=\frac{\sqrt{10}}{5}\)

- mathmate

just a recap,
we were working on simplify 9sqrt(1/3).
I sent this previous example to help.

- pooja195

sqrt 3/ 3

- mathmate

9 sqrt(1/3) = 9(sqrt(1)/sqrt(3)) = 9(sqrt(1)*sqrt(3)/(sqrt(3)^2)=9sqrt(3)/3=3sqrt(3)

- mathmate

or
\(9\sqrt{1/3}=3\frac{\sqrt3 \times \sqrt3}{\sqrt3}=3\sqrt3\)

- pooja195

IM IRGHT??

- pooja195

*RIGHT?

- mathmate

no, you had the 3 as denominator, and not multiplied.

- pooja195

>:(

- pooja195

9

- pooja195

.-.

- mathmate

not 9, it's \(3\sqrt3\) :)

- pooja195

>:(

- mathmate

ok, try this one:
Simplify \(\Large \sqrt{\frac{25}{3}}\)

- pooja195

\[\huge~\frac{ 5\sqrt{3} }{ 3 }\]

- pooja195

Am i right?

- pooja195

>:(

- mathmate

Excellent! It's correct!

- pooja195

:O YAY!!! :D

- mathmate

Yep! any questions before we move on?

- pooja195

Nope

- mathmate

Graphing parabolas (quadratic functions)

- pooja195

No we ahvent learned this

- mathmate

We need to be able to find the vertex of a parabola.

- pooja195

,-,

- mathmate

no?

- pooja195

^no

- mathmate

ok, next case!

- pooja195

kk

- mathmate

Do you know the shape of a quadratic function? (concave up, concave down)

- mathmate

When the leading coefficient is positive, it's concave up.

- pooja195

We havent learned this

- mathmate

|dw:1433641575278:dw|
Here coef. of x^2=1 (positive), so it is drawn concave up.

- mathmate

sure?

- pooja195

yea im sure .-.

- mathmate

ok, next case!

- mathmate

Solving quadratic equations by graphing, guess not learned!

- pooja195

^exactly

- mathmate

Using a graphics calculator?

- pooja195

nopee

- mathmate

Quadratic formula, no?

- mathmate

Si!

- pooja195

:3

- pooja195

lets avoid that ;)

- pooja195

Next case!

- mathmate

because you already know it, or because you don't?

- pooja195

because i know it :P

- pooja195

but i get the feeling yr gonna end up giving a problem anyways :P

- mathmate

Right!
I believe you, but I am sure you can solve this in a flash!

- mathmate

You know me toooooooo well!

- mathmate

Use the quadratic formula to solve the equation
2x^2+x-10=0

- mathmate

show work!

- mathmate

El cheapo, no LaTeX

- pooja195

i cant do it without latex

- mathmate

ok then!

- pooja195

a=2
b=1
c=-1
\[\huge~x=\frac{ -(1) \pm \sqrt{(1)^2-4(2)(-1)} }{ 2(2) } \]
\[\huge~x=\frac{ 1 \pm \sqrt{(9} }{ 4 } \]

- pooja195

Right so far?

- pooja195

\[\huge~x=\frac{ -1 \pm \sqrt{9\times 1} }{ 4 } \]

- pooja195

i meant -1 on the top sorry!

- mathmate

But you have -1 on top!
Can you simplify further?

- pooja195

\[\huge~x=\frac{-1 \pm 3\sqrt{1} }{ 4 }\]

- pooja195

Set up two of the equations + and - and then i get
x=0.5
x=−1

- mathmate

sorry, the question was:
2x^2+x-10=0
Can you make some adjustments?
such as :
\(\huge~x=\frac{-1 \pm 3\sqrt{1-4(2)(-10)} }{ 4 } = ...\)

- pooja195

OMG NO T_T

- pooja195

cant i just link u to a one ive done today T_T

- mathmate

ok, I believe you, I have seen you work before!

- pooja195

Thank you! :D

- mathmate

ok, discriminants!

- mathmate

Please tell me how many roots does this equation have:
-x^2+2x-1=0

- pooja195

2^2−4(−1)(1)=8
2 roots

- mathmate

Excellent!
how about this one?
\(2x^2-2x+3=0\)

- pooja195

(−2)^2−4(2)(3)=−20
no solution OR 2 imaginary roots

- mathmate

Excellent!
now this one, you can almost answer without looking:
\(x^2+2x+1=0\)

- pooja195

2 equal solutions :P

- mathmate

The answer (I mean guess) is correct.
Please justify your response!

- pooja195

-.-

- pooja195

2^2−4(1)(1)=0

- mathmate

Excellent. I would also guess the same answer, because there are only three possible cases! lol
Your books calls it "one solution", but I like yours better, "two equal solutions", or better still, two "coincident roots". So use any appropriate one of your choice.

- pooja195

kk

- mathmate

Graphing quadratic inequalities?

- pooja195

no

- mathmate

|dw:1433643467187:dw|

- pooja195

we havent learned it

- mathmate

still no?

- mathmate

ok, next case!

- mathmate

...more like next chapter!

- pooja195

lol :)

- mathmate

Chapter 10 Polynomials and factoring
========================

- mathmate

x^2 is a monomial
(x+2) is a binomial
(x^2+3x-1) is a trinomial.
Is x a polynomial?

- pooja195

no

- mathmate

"ERROR", the computer game says!

- pooja195

O_o

- mathmate

A polynomial includes all degrees, including x (first degree, one term only).

- mathmate

What is the degree of the polynomial 4.

- pooja195

linear/

- mathmate

linear is like 2x, 5x, which are abbreviations of \(2x^1, 5x^1\)

- pooja195

in that case i havent learned this ..-..

- mathmate

\(4=4x^0\), so the degree is zero!
(see p.569), perhaps the teacher went through it fast.

- pooja195

hmm ok

- mathmate

Do you know how to add, subtract, multiply and divide polynomials?

- pooja195

maybe >:) maybe not

- mathmate

We'll see!

- pooja195

of course

- mathmate

add:
\((x^2-7)-(x+2)=?\)

- mathmate

*subtract

- pooja195

\[x^2-7-x-2\]
\[x^2-x-9\]

- mathmate

Excellent!
now|dw:1433644413460:dw|
Find the perimeter and area of the swimming pool shown.

- pooja195

.-.not this

- pooja195

;-;

- mathmate

You find it hard, or too easy?

- pooja195

4x+3+4x+3+x-2+x-2
8x^2+2x+2

- pooja195

Next case!

- mathmate

sorry, not quite yet!

- pooja195

T_T

- mathmate

add like terms
perimeter
=4x+3+4x+3+x-2+x-2
=4x+4x+x+x +3+3-2-2
=10x+2

- mathmate

|dw:1433644752391:dw|

- pooja195

is that a question?

- mathmate

The first post was the corrected calculation for the perimeter.
The second post is to calculate the area using FOIL, to be completed by you!

- pooja195

4x^2−5x−6

- mathmate

Excellent!
For a rectangle:
perimeter = 2 times the sum of two adjacent sides.
Area = product of two adjacent sides.

- pooja195

ok

- mathmate

Have you done division?
(called long division, or synthetic division).

- pooja195

no

- mathmate

Like
Divide \(20x^2+17x+3\) by 5x+3.

- pooja195

no

- mathmate

ok, but can you do it using chapter 11?

- pooja195

omg no i hate that chapter >::( no no no ;(

- mathmate

\(\Large \frac{20x^2+17x+3}{5x+3}\)

- mathmate

But you were good at it, if I remember right!

- pooja195

I dont like that stuff T_T princess says skip

- mathmate

Teacher says "you pay now, or you pay later!" xD

- pooja195

T_T

- pooja195

skip

- mathmate

ok, for now!

- pooja195

>:) princess wins!

- mathmate

as always!

- mathmate

Now back to multiplication!
\((m^2+2m-9)(m-4)= \ ?\)

- pooja195

\[\huge~m^3−2m^2−17m+36 \]

- mathmate

Excellent! You're as good as a calculator!

- pooja195

:)

- mathmate

Good so far?

- pooja195

yes next case!

- mathmate

10.3 special products of polynomials
========================

- mathmate

- mathmate

- pooja195

+_=

- mathmate

gimme a minute to find the page

- mathmate

ok, special products of polynomials.
First, if you can, commit to memory the following:
Factoring difference of two squares
(a+b)(a-b)=a^2-b^2
Example:
x^2-y^2=(x+y)(x-y)

- mathmate

But life is not always simple like that.
the problem may come up as:
4p^2-9q^2

- pooja195

2p+3)(2p-3)

- mathmate

You will have to _try_ to factorize each term into a perfect square before proceeding,
4p^2-9q^2 = (2p)^2-(3q)^2 (remember the law of exponents?

- mathmate

Yes, pretty close, you only lost your q..lol

- pooja195

oops

- mathmate

(2p+3q)(2p-3q)

- mathmate

Expand
(a+2b)(a-2b)

- pooja195

a^2−4b^2

- mathmate

Excellent! You get the idea, and I'm sure you can do it backwards.

- mathmate

like
factor (16a^2-4b^2 ).

- pooja195

(4a+2b)(4a-2b)

- mathmate

Very good!
Now we move on to the perfect squares.
Commit to memory the following patterns:
(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2

- mathmate

Example: (backwards of FOIL)
9x^2-24x+16 = (3x)^2 - 2 (3x)(4) + (4^2) = (3x-4)^2

- mathmate

Note that perfect squares have three terms, and both the square terms are ALWAYS positive.
The middle term may be positive or negative, depending on the pattern.

- mathmate

I'll give you a couple of problems, but I have to go after that, perhaps for 50 minutes.
factor
1. 9x^2+30x+25
2. 4x^2-12x+9

- pooja195

:D ok!

- mathmate

gtg, but hope you have the answers by the time I'm back!

- pooja195

ok :)

- pooja195

\(\huge\color{blue}{1.~~ (3x+5)(3x+5)}\)
\(\huge\color{blue}{2.~~(2x−3)(2x−3) }\)

- mathmate

@pooja195
Yes, excellent.
I would have written them as squres, so that it is easier to read.
\( \huge\color{blue}{1.~~ (3x+5)^2}\)
\(\huge\color{blue}{2.~~(2x−3)^2 }\)

- mathmate

* squares

- mathmate

No you don't have to do foil to get the answer, but it's a good idea to do FOIL to _check_ the answer.

- mathmate

If there are no questions on 11.3, then next case.
11.4 ZERO PRODUCT PROPERTY
Let a and b be real number. If ab=0, then a=0 or b=0.
If the product of two factors is zero, then at least one of the factors must be zero.

- pooja195

thats eaasy

- mathmate

So if (x-3)(2x-5)=0, what are the possible values of x?

- pooja195

x=3
x=5/2

- mathmate

Excellent!
Solve by factoring:
\(\Large x^2-x-3=0\)

- pooja195

Cant do it .-.

- mathmate

sorry, it:
\(\Large x^2-x-6=0\)

- pooja195

x=−2 or x=3

- mathmate

yes! perfect!

- pooja195

Are we done? :D

- mathmate

no

- mathmate

brb

- mathmate

sorry!

- mathmate

Do you need more practice on factoring a quadratic?

- pooja195

No

- mathmate

...apart from word problems?

- pooja195

No i think i have it all

- mathmate

Just to make sure, try factor \(2x^2-9x-35\)

- pooja195

(2x+5)(x−7)

- mathmate

Why, that's too fast!?
Are you sure of your answer?

- pooja195

(2x+5)=0
(x−7)=0
x=-5/2
x=7

- mathmate

ok, now factor 3p^2+36p+108

- mathmate

Do you remember the first step in factoring?

- pooja195

GCF

- pooja195

in this case its 3

- mathmate

Excellent!!!!!!!

- pooja195

3(p+6)(p+6)

- mathmate

faster than I can type!
Is the calculator there somewhere? lol

- pooja195

maybe ;) but not for the whole problem :P

- mathmate

xD

- pooja195

How did u know ? :P

- pooja195

too fast? :P xD

- pooja195

i didnt use it for the whole thing just to find the factors :/

- mathmate

Aren't you on your laptop with a camera?

- pooja195

xD yesh :P

- mathmate

xD

- mathmate

next:
factor \(\large 4x^3+20x^2+24x\)

- pooja195

4x(x+2)(x+3)

- mathmate

Excellent!

- mathmate

I like it when you actually sweat it out!

- pooja195

xD

- mathmate

Now real cubic factors:
\((x+y)^3 = (x+y)(x^2-xy+y^2)\)
\((x-y)^3 = (x-y)(x^2+xy+y^2)\)
I'll sho you the SOAP rule Nnesha taught me today!

- pooja195

i havent learned this ,-,

- mathmate

Sorry, wrong formulas:
\(x^3+y^3 = (x+y)(x^2-xy+y^2)\)
\(x^3-y^3 = (x-y)(x^2+xy+y^2)\)

- mathmate

Sure?

- pooja195

yes

- mathmate

ok, then Ch 10 is done!

- pooja195

O_O

- pooja195

lets learn SOAP

- mathmate

It has to do with the cubic factoring, you really want it?

- pooja195

yes yes

- mathmate

k, gimme a minute.

- mathmate

|dw:1433700122466:dw|

- mathmate

The same SOAP rule applies to x^3-y^3 and x^3+y^3.

- mathmate

It's all about signs only.

- pooja195

oooo ok

- mathmate

Pretty cute, isn't it?

- pooja195

yesh :3

- mathmate

So, shall we start Ch 11?

- pooja195

Maybe we should do more quadratic factoring

- mathmate

ok!

- mathmate

Find the product
\(\large (d+2)(d^2-3d-10)\)

- pooja195

\[\huge~d^3−d^2−16d−20\]

- mathmate

Exactly!
Have you done the grid method of multiplication?

- pooja195

nope

- mathmate

It makes your life easier, even without a calculator!

- pooja195

O-o

- mathmate

Here's how it works. Nothing magical, just helps you organize.

- mathmate

|dw:1433700861581:dw|

- pooja195

ooo thats neat :o

- mathmate

Yep!

- pooja195

whats next? :)

- mathmate

find product:
(3x^3-5z^2+8)(z+2)

- mathmate

Be careful if you use the grid method, and treat the first factor as:
(3z^3-5z^2+0+8) to fill the gap.
and the first term is 3z^3, not 3x^3, sorry.

- pooja195

3x^3z+6x^3−5z^3−10z^2+8z+16

- mathmate

and simplify...

- mathmate

You mean 3x^4 at the beginning. :)

- pooja195

Right..

- mathmate

How about
factor \(\large 30x^2+38x+12\)

- pooja195

2(5x+3)(3x+2)

- mathmate

Very good!

- mathmate

Last one for chapter 10 (if you get it without calculator)
Factor \(\large 4x^2+44x+121\)

- pooja195

(2x+11)(2x+11)

- mathmate

Yep, excellent.

- mathmate

Unless there are questions, that's it for Ch 10.

- pooja195

nope thats all :P

- pooja195

mm chapter 11

- mathmate

11.1 Proportions
==========

- mathmate

solve \(\Large \frac{3}{y}=\frac{5}{8}\)

- mathmate

brb

- pooja195

8*3=24
5y
5y=24
y=24/5

- mathmate

Yep, that's good. That's cross multiplication.

- mathmate

We can extend the idea...

- mathmate

Solve \(\Large \frac{2}{x-3}=\frac{7}{x+2}\)

- pooja195

2(x+2) = 7(x-3)
2x+4=7x-21
2x=7x-25
-5x=-25
x=5

- mathmate

Excellent.
Now try solve \(\Large \frac{x}{x-4}=\frac{6x}{x+1}\)

- pooja195

x(x+1)=6x(x-4)
x^2+1x=6x^2-24x
-7x^2+1x=-24x
not sure...

- mathmate

x^2+1x=6x^2-24x is good, work from here.

- pooja195

idkk :/

- mathmate

x^2+1x=6x^2-24x
subtract x^2+x on each side:
x^2+x - (x^2-x) = 6x^2-24x -x^2 -x
0 = 5x^2-25x
0=5x(x-5)
so x=0 or x=5

- pooja195

:/

- mathmate

Solve \(\Large \frac{5}{x+2}=\frac{3x-1}{x^2-1}\)

- pooja195

5(x^2-1) = (3x-1)(x+2)
5x^2−5=3x^2+5x−2
5x^2−5−(3x^2+5x−2)=3x^2+5x−2−(3x^2+5x−2)
2x^2−5x−3=0
(2x+1)(x−3)=0
2x+1=0 or x−3=0
x= -1/2 or x=3

- mathmate

Excellent, like a pro!

- mathmate

Any question on proportions?

- pooja195

nope

- mathmate

11.2 Direct and inverse variations.
=====================

- mathmate

It's important to know what's what.

- mathmate

Can you give me an example of one of each?

- mathmate

Usually it's in the form y=.....

- mathmate

Direct variation: y=kx, example y=x, y=x/2, y=3.2x
Inverse variation: xy=k, or y=k/x Example y=4/x, or xy=10

- mathmate

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