pooja195 one year ago @mathmate

1. pooja195

@mathmate

2. pooja195

Chapter 9 ;-;

3. mathmate

Chapter 9 ======

4. mathmate

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

5. pooja195

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

6. mathmate

Yep, now evaluate x numerically, please.

7. pooja195

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

8. mathmate

So x=13 (is the final answer).

9. mathmate

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

10. pooja195

idk this ;-;

11. 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?

12. pooja195

no .-.

13. pooja195

*none

14. 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?

15. pooja195

Decimals? .-.

16. pooja195

wait no

17. mathmate

no, we are looking for two integer numbers! lol

18. pooja195

This si confusing T_T lets skip this

19. 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!

20. pooja195

-_-

21. 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?

22. pooja195

yes

23. mathmate

Solve the equation 27-3y^2=0

24. pooja195

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

25. mathmate

put in the work, but LaTeX is not required.

26. 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

27. pooja195

/.\

28. 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=$$\pm 3$$

29. mathmate

30. 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.

31. pooja195

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

32. mathmate

Take out your ti-83 to finish!

33. mathmate

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

34. mathmate

sqrt(1.375)=?

35. pooja195

1.172604

36. mathmate

exactly! (1.17 for 2 decimal places).

37. mathmate

Any questions before we move to simplifying radicals?

38. pooja195

no

39. 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.

40. mathmate

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

41. mathmate

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

42. pooja195

48 6 8 3 2 42 $2\sqrt{12}$

43. mathmate

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

44. mathmate

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

45. mathmate

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

46. pooja195

Dance club.

47. pooja195

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

48. mathmate

Oh! Can you show me that?

49. 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

50. 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)

51. pooja195

lel :3

52. pooja195

Are we done with this section ?

53. 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.

54. pooja195

T_T

55. pooja195

ok

56. mathmate

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

57. pooja195

im confused whats the question? .-.

58. mathmate

simplify sqrt(60)

59. pooja195

60 10 6 5 2 3 2 2 sqrt 15

60. pooja195

T_T say im right!

61. pooja195

Princess demands it!

62. mathmate

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

63. mathmate

Now we move onto rationalizing the denominator.

64. mathmate

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

65. pooja195

Just multiply by the root

66. pooja195

*in the denominator

67. mathmate

68. pooja195

Pretest

69. mathmate

Rationalize 9sqrt(1/3)

70. pooja195

9 squrt 3

71. pooja195

^LOL

72. mathmate

Sure?

73. mathmate

The question was $$9\sqrt{\frac{1}{3}}$$

74. pooja195

:/

75. pooja195

this is diffrent .-.

76. mathmate

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

77. mathmate

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

78. pooja195

sqrt 3/ 3

79. mathmate

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

80. mathmate

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

81. pooja195

IM IRGHT??

82. pooja195

*RIGHT?

83. mathmate

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

84. pooja195

>:(

85. pooja195

9

86. pooja195

.-.

87. mathmate

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

88. pooja195

>:(

89. mathmate

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

90. pooja195

$\huge~\frac{ 5\sqrt{3} }{ 3 }$

91. pooja195

Am i right?

92. pooja195

>:(

93. mathmate

Excellent! It's correct!

94. pooja195

:O YAY!!! :D

95. mathmate

Yep! any questions before we move on?

96. pooja195

Nope

97. mathmate

98. pooja195

No we ahvent learned this

99. mathmate

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

100. pooja195

,-,

101. mathmate

no?

102. pooja195

^no

103. mathmate

ok, next case!

104. pooja195

kk

105. mathmate

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

106. mathmate

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

107. pooja195

We havent learned this

108. mathmate

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

109. mathmate

sure?

110. pooja195

yea im sure .-.

111. mathmate

ok, next case!

112. mathmate

Solving quadratic equations by graphing, guess not learned!

113. pooja195

^exactly

114. mathmate

Using a graphics calculator?

115. pooja195

nopee

116. mathmate

117. mathmate

Si!

118. pooja195

:3

119. pooja195

lets avoid that ;)

120. pooja195

Next case!

121. mathmate

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

122. pooja195

because i know it :P

123. pooja195

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

124. mathmate

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

125. mathmate

You know me toooooooo well!

126. mathmate

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

127. mathmate

show work!

128. mathmate

El cheapo, no LaTeX

129. pooja195

i cant do it without latex

130. mathmate

ok then!

131. pooja195

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

132. pooja195

Right so far?

133. pooja195

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

134. pooja195

i meant -1 on the top sorry!

135. mathmate

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

136. pooja195

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

137. pooja195

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

138. mathmate

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

139. pooja195

OMG NO T_T

140. pooja195

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

141. mathmate

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

142. pooja195

Thank you! :D

143. mathmate

ok, discriminants!

144. mathmate

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

145. pooja195

2^2−4(−1)(1)=8 2 roots

146. mathmate

Excellent! how about this one? $$2x^2-2x+3=0$$

147. pooja195

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

148. mathmate

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

149. pooja195

2 equal solutions :P

150. mathmate

151. pooja195

-.-

152. pooja195

2^2−4(1)(1)=0

153. 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.

154. pooja195

kk

155. mathmate

156. pooja195

no

157. mathmate

|dw:1433643467187:dw|

158. pooja195

we havent learned it

159. mathmate

still no?

160. mathmate

ok, next case!

161. mathmate

...more like next chapter!

162. pooja195

lol :)

163. mathmate

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

164. mathmate

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

165. pooja195

no

166. mathmate

"ERROR", the computer game says!

167. pooja195

O_o

168. mathmate

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

169. mathmate

What is the degree of the polynomial 4.

170. pooja195

linear/

171. mathmate

linear is like 2x, 5x, which are abbreviations of $$2x^1, 5x^1$$

172. pooja195

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

173. mathmate

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

174. pooja195

hmm ok

175. mathmate

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

176. pooja195

maybe >:) maybe not

177. mathmate

We'll see!

178. pooja195

of course

179. mathmate

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

180. mathmate

*subtract

181. pooja195

$x^2-7-x-2$ $x^2-x-9$

182. mathmate

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

183. pooja195

.-.not this

184. pooja195

;-;

185. mathmate

You find it hard, or too easy?

186. pooja195

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

187. pooja195

Next case!

188. mathmate

sorry, not quite yet!

189. pooja195

T_T

190. mathmate

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

191. mathmate

|dw:1433644752391:dw|

192. pooja195

is that a question?

193. 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!

194. pooja195

4x^2−5x−6

195. mathmate

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

196. pooja195

ok

197. mathmate

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

198. pooja195

no

199. mathmate

Like Divide $$20x^2+17x+3$$ by 5x+3.

200. pooja195

no

201. mathmate

ok, but can you do it using chapter 11?

202. pooja195

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

203. mathmate

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

204. mathmate

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

205. pooja195

I dont like that stuff T_T princess says skip

206. mathmate

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

207. pooja195

T_T

208. pooja195

skip

209. mathmate

ok, for now!

210. pooja195

>:) princess wins!

211. mathmate

as always!

212. mathmate

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

213. pooja195

$\huge~m^3−2m^2−17m+36$

214. mathmate

Excellent! You're as good as a calculator!

215. pooja195

:)

216. mathmate

Good so far?

217. pooja195

yes next case!

218. mathmate

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

219. mathmate

@pooja195

220. mathmate

@pooja195

221. pooja195

+_=

222. mathmate

gimme a minute to find the page

223. 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)

224. mathmate

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

225. pooja195

2p+3)(2p-3)

226. 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?

227. mathmate

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

228. pooja195

oops

229. mathmate

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

230. mathmate

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

231. pooja195

a^2−4b^2

232. mathmate

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

233. mathmate

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

234. pooja195

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

235. 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

236. mathmate

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

237. 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.

238. 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

239. pooja195

:D ok!

240. mathmate

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

241. pooja195

ok :)

242. pooja195

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

243. mathmate

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

244. mathmate

* squares

245. 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.

246. 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.

247. pooja195

thats eaasy

248. mathmate

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

249. pooja195

x=3 x=5/2

250. mathmate

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

251. pooja195

Cant do it .-.

252. mathmate

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

253. pooja195

x=−2 or x=3

254. mathmate

yes! perfect!

255. pooja195

Are we done? :D

256. mathmate

no

257. mathmate

brb

258. mathmate

sorry!

259. mathmate

Do you need more practice on factoring a quadratic?

260. pooja195

No

261. mathmate

...apart from word problems?

262. pooja195

No i think i have it all

263. mathmate

Just to make sure, try factor $$2x^2-9x-35$$

264. pooja195

(2x+5)(x−7)

265. mathmate

266. pooja195

(2x+5)=0 (x−7)=0 x=-5/2 x=7

267. mathmate

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

268. mathmate

Do you remember the first step in factoring?

269. pooja195

GCF

270. pooja195

in this case its 3

271. mathmate

Excellent!!!!!!!

272. pooja195

3(p+6)(p+6)

273. mathmate

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

274. pooja195

maybe ;) but not for the whole problem :P

275. mathmate

xD

276. pooja195

How did u know ? :P

277. pooja195

too fast? :P xD

278. pooja195

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

279. mathmate

Aren't you on your laptop with a camera?

280. pooja195

xD yesh :P

281. mathmate

xD

282. mathmate

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

283. pooja195

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

284. mathmate

Excellent!

285. mathmate

I like it when you actually sweat it out!

286. pooja195

xD

287. 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!

288. pooja195

i havent learned this ,-,

289. mathmate

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

290. mathmate

Sure?

291. pooja195

yes

292. mathmate

ok, then Ch 10 is done!

293. pooja195

O_O

294. pooja195

lets learn SOAP

295. mathmate

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

296. pooja195

yes yes

297. mathmate

k, gimme a minute.

298. mathmate

|dw:1433700122466:dw|

299. mathmate

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

300. mathmate

301. pooja195

oooo ok

302. mathmate

Pretty cute, isn't it?

303. pooja195

yesh :3

304. mathmate

So, shall we start Ch 11?

305. pooja195

Maybe we should do more quadratic factoring

306. mathmate

ok!

307. mathmate

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

308. pooja195

$\huge~d^3−d^2−16d−20$

309. mathmate

Exactly! Have you done the grid method of multiplication?

310. pooja195

nope

311. mathmate

It makes your life easier, even without a calculator!

312. pooja195

O-o

313. mathmate

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

314. mathmate

|dw:1433700861581:dw|

315. pooja195

ooo thats neat :o

316. mathmate

Yep!

317. pooja195

whats next? :)

318. mathmate

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

319. 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.

320. pooja195

3x^3z+6x^3−5z^3−10z^2+8z+16

321. mathmate

and simplify...

322. mathmate

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

323. pooja195

Right..

324. mathmate

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

325. pooja195

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

326. mathmate

Very good!

327. mathmate

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

328. pooja195

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

329. mathmate

Yep, excellent.

330. mathmate

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

331. pooja195

nope thats all :P

332. pooja195

mm chapter 11

333. mathmate

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

334. mathmate

solve $$\Large \frac{3}{y}=\frac{5}{8}$$

335. mathmate

brb

336. pooja195

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

337. mathmate

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

338. mathmate

We can extend the idea...

339. mathmate

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

340. pooja195

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

341. mathmate

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

342. pooja195

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

343. mathmate

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

344. pooja195

idkk :/

345. 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

346. pooja195

:/

347. mathmate

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

348. 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

349. mathmate

Excellent, like a pro!

350. mathmate

Any question on proportions?

351. pooja195

nope

352. mathmate

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

353. mathmate

It's important to know what's what.

354. mathmate

Can you give me an example of one of each?

355. mathmate

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

356. 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

357. mathmate

|dw:1433706731769:dw|

358. mathmate

|dw:1433706794615:dw|

359. pooja195

Direct variation: rule y=kx (when x is greater, y is greater) Inverse variation: rule xy=k (when x increases, y decreases, that's why inverse) right? This is from the other post.

360. mathmate

That's true, exactly!

361. mathmate

So if I give you: x y 1 3 2 6 3 9 Is this direct or inverse variation?

362. pooja195

>_< can we not do these .-.

363. mathmate

These are needed to solve word problems!

364. pooja195

O_O

365. mathmate

Say x=4/y, direct or inverse?

366. mathmate

Ok, I'll put it another way. When the function is in the slope-intercept form, with $$b=0$$, then it is direct variation.

367. mathmate

if it is something to do with xy=something, it is inverse.

368. mathmate

If it is in slope-intercept with intercept NOT equal to zero, it's partial (neither direct, nor inverse)