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Matte 2 - Algebra och ickelinjära modeller


Andragradsekvationer

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Definition

Andragradsekvationer kallas de ekvationer där variabeln är upphöjd till 2 t.ex. . Om variabeln hade varit upphöjd till 3 så hade det varit en tredjegradsekvation osv. Man använder alltså samma namngivning som för polynom av olika grader.

En ekvation har alltid lika många lösningar som dess grad anger. Med andra ord, en andragradsekvation har alltid 2 lösningar!

En andragradsekvation kan allmänt skrivas där a, b och c är konstanter och . Om a hade varit lika med noll så hade det ju inte funnits någon x2 och därmed hade vi inte haft någon andragradsekvation.

Det finns flera olika metoder för att lösa andragradsekvationer, vissa funkar alltid, andra funkar bara för specialfall.

Kvadratrotsmetoden

M h a kvadratrotsmetoden använder vi “kvadratroten ur” för att lösa ekvationer som enbart utgörs av variabeln upphöjt till två t.ex. .
Då vi tar kvadratroten ur ett tal så får vi inte glömma att skriva så här:

Som vi sa nyss, eftersom det är en andragradsekvation så har den två lösningar men vi ser bara en om vi inte sätter dit plus-minus-tecknet. Då vi kvadrerar ett negativt tal blir det ju positivt. Man kan dock aldrig ta roten ur ett negativt tal, sådana saknar lösning.

Denna andragradsekvation har de båda lösningarna och dvs. +7 och -7.

Nollproduktmetoden

Nollproduktmetoden går ut på att om en produkt är 0 så måste minst en utav faktorerna vara noll. Detta kan vi utnyttja i exemplet nedan:

Denna andragradsekvation kan vi faktorisera och därmed bryta ut x:

Vi ser nu att eftersom multiplikationen mellan x och (x+2) blir 0 så måste antingen x eller parentesen vara 0.
Första lösningen,
och andra lösningen fås genom

Nollproduktmetoden fungerar även för ekvationer med högre grad se exemplen nedan:

Exempel 1

Lös ekvationen:


Den här ekvationen består av tre faktorer: x, (x+4) och (x+3) och är då en tredjegradare, vilket innebär att den har tre lösningar. Eftersom produkten är noll innebär det att någon utav dessa tre faktorer är lika med noll. Då sätter vi helt enkelt varje faktor lika med noll, vilket ger oss lösningarna:


Svar:

Exempel 2

Lös ekvationen:


Här samlar vi först alla x på ena sidan och bryter sedan ut x-termen, alltså gör vi en faktoruppdelning/faktorisering.

Även här så måste någon av faktorerna vara lika med noll då produkten är noll.

Svar:

Kvadratkomplettering

Låt säga att vi har andragradsekvationen . Den kan inte lösas med någon av de metoderna som nämnts ovan utan här krävs det att vi använder kvadratkomplettering.

Kvadratkompletteringen går ut på att vi “kompletterar” vår ekvation med ett tal i kvadrat så att vänsterledet får utseendet som ett av de polynom vi associerar med kvadreringsreglerna. Vi tittar på ekvationen igen: – Vad är det för term som saknas för att vi ska kunna skriva om vänsterled m h a kvadreringsreglerna? Vi ser att både och är positiva alltså får vi anpassa oss till utseendet som fås via första kvadreringsregeln.

Jämförelse:
Förklaring till kvadratkomplettering

Vi ser att b2 ska motsvaras av 22, vilket vi sätter in i vår ekvation. Kom ihåg att det vi gör på ena sidan om likhetstecknet måste vi även göra på andra sidan likhetstecknet. Alltså sätter vi in även i högerled.

I och med detta kan vi skriva om vänsterled till enligt första kvadreringsregeln samtidigt som högerled bara beräknas:

Nästa steg i ekvationslösningen är att ta bort kvadreringen. Även detta måste göras i båda leden. För att ta bort kvadreringen tar vi roten ur då upphöjt till 2 och roten ur tar ut varandra. Vi tar även roten ur 16.


Om vi testar respektive x-värde genom att sätta in det i ursprungsekvationen så ser vi att det stämmer;

Exempel 3

Lös ekvationen med kvadratkomplettering.



Vi börjar med att skriva om ekvationen till . Därefter behöver vi titta på vad vi ska komplettera med för kvadrat för att få ett uttryck som är rätt skrivet efter kvadreringsreglerna. I vänsterled ser vi som skulle motsvara:
enligt andra kvadreringsregeln. Vi inser att frågetecknet ska ersättas med siffran 1, alltså hälften av den siffra som står framför x-termen.

Då detta är en ekvation vi jobbar med så måste vi även kvadrera i högerled:

Enligt andra kvadreringsregeln motsvarar detta nu:

Nu tar vi bort kvadreringen genom att ta roten ur VL, detta måste även göras i HL:


Vi prövar lösningarna genom att sätta in dem i ekvationen var för sig:

Det stämmer! :D

Svar: x = 5 och x = -3

PQ-formeln

PQ-formeln är snabblösningen på det problem vi ställs inför när en andragradsekvation måste lösas med kvadratkomplettering. Det är en formel som är framtagen för att man snabbt och enkelt ska kunna få ut x-värdena utan att man själv behöver kvadratkomplettera.

PQ-formeln med lösning:


där p och q är konstanter. x² + px + q = 0 är den karakteristiska ekvationen för att lösningen på x ska bli möjlig.
Till exempel så måste x² alltså stå “ensamt”. Det får alltså inte vara 2x² + 2 + 4 = 0, för då fungerar inte pq formeln. Detta kan man dock enkelt lösa genom att dividera alla led med 2 i detta exempel, för att sedan kunna använda formeln.

Exempel 4

Lös ekvationerna med hjälp av pq formeln:
a.)
b.)
c.)


a.)

b.)

c.)

Rotekvationer

Rotekvationer kallas sådana ekvationer som har det obekanta talet under rottecknet. Då man löser rotekvationer måste man kvadrera, vilket bidrar till att den nybildade ekvationen inte är lika med den förra ekvationen. Detta medför, då ekvationen är löst, att man har en falsk rot. Den falska roten upptäcks genom prövning, man testar om svaret är rätt genom att sätta in det i den ursprungliga ekvationen.

Exempel 5

Lös ekvationen:


Börja med en förenkling:

Låt rottecknet stå själv på ena sidan och kvadrera sedan talen på båda sidorna. x-1 fås av att ‘roten ur’ är samma sak som ‘upphöjt i 0,5′ och när vi sedan kvadrerar så blir det en multiplikation av potenserna 0,5*2 = 1, alltså Notera också att vi använder pilenvid kvadreringen. Den betyder “medför att”.
Nu gör vi om ekvationen så vi kan lösa den med:

Nu ska vi avslöja den falska roten genom att testa de båda svaren i den ursprungliga ekvationen
Prövning ger:


HL = 5, VL = HL.


HL = 2, VL HL

(VL = Vänster led, HL = Höger led)

Svar:

Lösa rotekvationer med substitution

Substitution betyder i detta fall ”ersätta”, vilket är det perfekta ordet för det vi just nu ska göra. För att förklara denna metod behövs ett exempel.

Exempel 6

Vi ska lösa ekvationen:


Om vi istället säger att där (vi ersätter ‘roten ur x’ med ‘u’) så får vi automatiskt att
Ursprungsekvationen ser då ut så här istället:

Om vi sedan flyttar över sexan till andra sidan av likhetstecknet så har vi hux flux en andragradsekvation som vi lätt kan lösa mha PQ-formlen! :D



ger oss eftersom
duger inte eftersom vi ställde villkoret att u skulle vara större än noll i början.

Svar:
Obs! Mycket viktigt att komma ihåg att svara med x och inte u, vilket många tenderar att glömma!




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Äldre kommentarer

  1. Sheryl Ilao Åström
    18 november 2015 @ 11:08

    Hej!

    Kom över denna sidan då jag letar bra exempel på rotekvationer. Såg dock exemplet med x^{2} 4x=12 här ovan och funderar varför det inte skulle gå att lösa den som alla andra andragradsekvationer genom att bara föra över 12 så det blir minus 12. Får ut 2 x som går att sätta in i den ursprungliga lösningen. x1=-6 och x2=2



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Watch Full HD1080p Blu-ray Disc movies, plus wirelessly stream a wide variety of online entertainment including movies, TV episodes, videos, music and live sports from Netflix, YouTube, HuluPlus, Pandora, and more with the BDP-S580 3D Blu-ray player with built-in Wi-Fi. You can also turn your iPhone, iPad or compatible Android phone into a versatile remote control when you download the free "Media Remote" app. Add compatible Sony Network speakers and instantly enjoy multi-room wireless audio streaming throughout your home.

From the Manufacturer

3D Technology Checklist

This product is 3D-related. To help you get a great 3D experience, use the checklist below to ensure you have everything you need. 3D viewing requires:

A Display
First, you'll need a 3D-ready display--whether it's a 3D HDTV, 3D projector, or 3D computer monitor. These displays have more processing power than standard 2D models for displaying 3D images in rapid succession. A Source
Your display may be ready for 3D playback, but you'll still need a device to read 3D content. This can be a cable box with a subscription to a 3D channel, a 3D Blu-ray Disc player, or a PlayStation 3 system. 3D Content
3D content--the actual entertainment, in other words--will be played back using the source mentioned above, whether it's a 3D broadcast from your cable provider, a 3D Blu-ray Disc, or a 3D video game. 3D Glasses
For now, the vast majority of 3D HDTVs require glasses for 3D viewing. Many use powered "active shutter" glasses, others polarized "passive" glasses. You'll need one pair per viewer, and they'll have to be compatible with your display, whether they're the same brand, or a pair of "universal" glasses designed to work across brands. HDMI Cable
To connect your source (such as a 3D Blu-ray Disc player) to your display, you'll need a high-speed HDMI cable. Cables with this designation feature bandwidth speeds up to 10.2 Gbps (gigabits per second), for carrying the 3D signal without any loss of quality.
If you want to get more information about 3D, shop our 3D products, watch videos, or interact with other customers, we invite you to visit 3D 101, our customer center about everything 3D.

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February 2011
Enjoy Full HD1080p Blu-ray Disc movies--even in 3D. Plus, connect to the Internet wirelessly out of the box, for streaming a wide variety of online entertainment, including movies, TV episodes, videos, music, and live sports from Netflix, YouTube, HuluPlus, Pandora and more with the BDP-S580 3D Blu-ray Disc player. You can also turn your iPhone, iPad or compatible Android phone into a versatile remote control with search functionality when you download the free "Media Remote" app. Add compatible Sony HomeShare network speakers for multi-room wireless audio streaming throughout your home.

Sony BDP-S580

Endless Possibilities


Sony BDP-S580 3D Get immersed in 3D Blu-ray movies in Full HD 1080p resolution. Sony BDP-S580 Streaming Instantly stream a wide variety of movies, TV shows, live sports, videos and music from Netflix, YouTube, Pandora, Hulu Plus and many more. Built-In Wi-Fi

Easily connect to the internet through your home's wireless broadband network thanks to built-in Wi-Fi. No need to purchase an adapter or other device, just fire it up, connect, and you'll have access to streaming content, DLNA content on your network, and more.

3D in Full HD 1080p

Feel like you've been transported into the action. A perfect complement to any home theater, the BDP-S580 is ready for the future of home entertainment. When paired with your 3D HDTV and active shutter glasses, you'll get the ultimate at-home 3D viewing experience.

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Streaming Content

With the BDP-S580 Blu-ray Disc player, there's always something on. Using your home broadband internet connection, you can instantly stream a wide variety of movies, TV shows, live sports, videos and music from Netflix, YouTube, Pandora, Hulu Plus and many more. Connect to your home network with a standard Ethernet cable or via built-in Wi-Fi.

Sony's I/P Noise Reduction is also on-board, helping to improve the picture quality of Internet entertainment and enjoy crisp, clear images on your HDTV--even with compressed, streaming media.

Sony BDP-S580 Remote Turn your iPhone, iPad, or compatible Android phone into a versatile remote when you download Sony's free "Media Remote" app. Sony BDP-S580 Wi-Fi Access network features wirelessly right out of the box with built-in Wi-Fi. Mobile Device Control

Turn your iPhone, iPad or compatible Android phone into a versatile remote when you download Sony's free "Media Remote" app. This gives you basic remote functions, a full QWERTY keyboard, plus the ability to search for information about the movies you're watching on YouTube, Wikipedia, and many BRAVIA Internet Video services. Better yet, share information with your friends via Twitter.

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Get your popcorn ready. With Quick Start/Quick Load, you will begin enjoying your entertainment faster than ever.

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Connect to the internet and get detailed information about the movie you're watching with Gracenote service. This easy-to-navigate service displays handy info such as cast information, genre, jacket art and more right on your TV screen. You can also access select internet apps including YouTube and Qriocity to find related entertainment.

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Hook up your digital camera, camcorder, USB flash drive, or USB-enabled MP3 player via the built-in USB input and view pictures, videos or listen to music on your TV.

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Support for Dolby TrueHD and dts-HD (Master Audio and High Resolution Audio) codecs means you can get cutting-edge, high-definition sound with 7.1 channels of discrete audio, for a truly cinematic experience.

Stream Audio from Room to Room

Do more than play your favorite movies. Outfitted with PartyStreaming technology, you can add Sony HomeShare network speakers (SA-NS400 and SA-NS300 models, sold separately), and then pop in your favorite CD or stream music from Qriocity, Pandora, Slacker, vTuner, or NPR to any room of your home.

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BDP-S580 Select Specifications Video Resolution: 1080p Playable Disc Types: CD-R, Blu-ray Disc, BD-RE, BD-R, BD-ROM, 3D BD-ROM, CD-DA, CD, DVD-Video, DVD-RW, DVD-R (Incl. Dual Layer), DVD+R (Incl. Double Layer), DVD+RW, CD-RW, BDAV, BD/CD, BD/DVD Playable File Types: AVCHD, MPEG 1, MPEG 2, MPEG 4, WMV9, AAC, WAV, MP3, WMA9, GIF, JPEG, PNG Inputs/Outputs HDMI Output 1 Composite Video Output: 1 Component Video Output: 1 Digital Audio Output 1 (Coaxial) Analog Audio Output 1 USB Input 2 (1 Front, 1 Rear) Ethernet Port: 1 Weight and Dimensions Dimensions (WxHxD): 17 x 1.4 x 7.8 Inches General Features Region Code (BD) : A Region Code (DVD): 1 Media Format: PAL, NTSC What's in the Box

Sony BDP-S580 Blu-Ray Disc Player (Black), Remote Control With Batteries, AV Cable, Documentation

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Technical Details

Brand Name Sony Item Weight 1.2 pounds Product Dimensions 7.8 x 17 x 1.4 inches Item model number BDPS580 Color Name Black Special Features can play video placed upside down., is a failproof product

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ASIN B004K1EOCA Customer Reviews 4.1 out of 5 stars 639 customer reviews
4.1 out of 5 stars Best Sellers Rank #80,632 in Electronics (See top 100)
#222 in Electronics > Televisions & Video > Blu-ray Players & Recorders > Blu-ray Players
Shipping Weight 5.3 pounds (View shipping rates and policies) Domestic Shipping Item can be shipped within U.S. International Shipping This item can be shipped to select countries outside of the U.S. Learn More Date First Available January 20, 2011

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There was a problem filtering reviews right now. Please try again later. 4.0 out of 5 stars Great Blu-Ray, on demand video serious problem By A. Brooks on March 20, 2011 Verified Purchase We have had excellent experiences with Sony video and audio products. Their devices cost a little more than their competitors but they are high quality, last a long time, and are easy to use. All of our electronics (except computers) are Sony. This blu-ray is no exception. Easy to setup, easy to use, some surprise features, quiet.

Well the problem is on demand video from netflix or amazon or any other service. It simply doesnt work... often. Its registered, and I verified it with Sony. Its connected to the internet with no problems and the browser works. Movies stream no problem on my PC. We have 6Mbs DSL so movies stream no problems on our PC. I even placed it in the DMZ to allow all ports to talk to it without restriction, no dice. Netflix has the unit registered and it is receiving data. It connected wired and wirelessly with neither working correctly with on demand video services.

We got 1 movie to work only. All the rest of the time it doesn't connect to the service or is only partially connected. The one time we were able to watch a movie it was flawless with no hiccups.

NOTE: If you want to watch blu-ray currently as of this posting you need 10mbs internet speed.

We are very very sad. We specifically bought this for Netflix and can't use it. This is why the unit gets 4 stars and not 5.

If anyone knows something we don't please let us know.

--- LATER

Well we found the problem. If you get an IP address for the unit from DHCP it will cause connection problems. Once we assigned the unit a static IP address (manual) with mask, default gateway, and both DNS (manual) the unit works flawlessly. DNS servers that worked: 8.8.8.8 and 8.8.4.4

---- 2 MONTHS LATER

For best speed and connectivity set the DNS to Google's DNS servers (8.8.8.8 and 8.8.4.4). Don't use your DSL/Cable Modem Router as DNS.

---- 18 MONTHS LATER Aug/2012

With the new updates for the Sony device we now rarely have a problem with movies not loading, in fact I cant remember the last time it didnt load. Occasionally a movie/show does hiccup on Netflix and stop. You just hit the "menu" button and restart it. Its probably a packet loss over the internet. We have cable modem so I am not sure if the same effect happens on DSL. Sony is constantly adding new channels to our list. We actually got rid of cable, $70 a month for basic cable, in favor of just Netflix, VUDU, and Amazon. Between those 3 you can get just about anything. Still love the product and no I dont work for Sony ;) Your welcome to all in which my post has helped.

*Angy's husband, Al.... in house tech support* Read more 0 Comment | 382 people found this helpful. Was this review helpful to you? Yes No Report abuse 4.0 out of 5 stars Very nice addition to my Sony Bravia home entertainment products By K. Hyde on December 25, 2011 Verified Purchase Given that I own a Sony Bravia TV, home theater system and iPod dock I wanted to finish the whole shabang off with a Sony Bravia blu ray player. I wanted to ensure that my master Sony remote would operate every Bravia product I own and this is a compatible product. I want one remote that controls it all and the all-in-ones I've seen don't support the entire Bravia Sync system.

In no way, shape or form am I a technical person so I'm speaking from very non-technical points as I do this review.

Positive points:

1. This works beautifully with the Belkin wireless router (Belkin N600 Wireless Dual-Band N+ Router (Latest Generation)) to allow me to stream videos from my Amazon Prime membership with no skipping or freezing. Amazon Instant Videos and Netflix applications are built into the player as are a few other video options such as Hulu Plus. I only have the most basic cable plan that provides about 12 stations, so having these other options available to me increases my viewing possiblilities without the ridiculous price of cable TV. If you're a documentary kind of person like me you'll find a bazillion things on Amazon Instant Video with Prime Membership to keep you well entertained.
2. It's also got music applications such as Pandora built in. There's another one called Slacker, but I find Pandora to be ideal for my music needs.
3. As a blu ray player it has a nice sharp picture. Coupled with my Sony home theater system it's one heck of a theater experience.
4. Amazon plug - easily the best price I found by at least $20.

Negative points:

1. I'm not sure that I could have installed this without the assistance of my incredibly smart 18 year old nephew and by no means am I a dolt. Finding the appropriate codes to activate certain applications was rather frustrating even with his assistance. One thing I've learned over the past 2 years of being a Sony Bravia component owner is that Sony makes great products, but writes lousy manuals.
2. When loading up a blu ray it is fairly noisy to the point that I wondered if I loaded something incorrectly or perhaps there was a defect to it. It also seems to take a while to load vs. the Sony Bravia Upscaling DVD player this replaced.
3. The internet application on it is about as handy as a doorknob on a tree - it's teensy tiny to read and not easy to navigate around. I cannot see any reason that I'll ever use this function other than when I first got it to see how it worked. Use your computer or iPad type of product instead if you want to look something up - this application will just frustrate you.

I cannot comment on its ability as a 3D player since I don't own a 3D TV, not to mention that it would make me incredibly motion sick. I already wear glasses and don't want to wear other glasses on top of them either.

If you're going to use this wirelessly be absolutely sure that you already own a router than can support it such as the Belkin model I mentioned earlier. There's one level of router above the one I purchased to use along with this, but that level isn't necessary unless you're having multiple wireless users in your home according to tech sources much more knowledgable than I. I've simultaneously streamed videos to this player while using my laptop on wi-fi with no difficulties whatsoever.

And as a plug for Amazon, their Prime Membership Instant Video library is improving all that time and will very shortly rival Netflix - it's not quite there yet, but still has a variety of movies, TV programs and documentaries to choose from. I did the math - it would be about $110/yr (including tax) for Netflix instant video and I wouldn't be getting 2 day shipping like I do with my $79/yr Prime Membership (which doesn't require the minimum $25 for the free shipping - you can get a $4.95 product shipped at no additional cost so long as Amazon or one of their retailers participates in the program). I know I sound like an ad for Amazon, but it's definitely something to consider if you don't already have a Netflix membership and want to get the most bang for your buck. Read more 0 Comment | 5 people found this helpful. Was this review helpful to you? Yes No Report abuse See all 639 reviews Write a customer review

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5.0 out of 5 stars Well designed Well designed Published 4 months ago by John Allen 5.0 out of 5 stars Five Stars Great price; great product Published 8 months ago by Stephen J. 5.0 out of 5 stars Five Stars perfect Published 11 months ago by Ronald Mateo 5.0 out of 5 stars Loved it so much bought 2 more just like it Loved it so much bought 2 more just like it. Does everything I could want in blueray and more. Published 11 months ago by john 5.0 out of 5 stars Super Audio CD decoder. AWESOME. Excellent blu Ray player. Still going strong in 2016. Composite audio outputs to run to a vintage receiver as well as HDMI. DECODES SUPER AUDIO MUSIC CD's. Published 11 months ago by BOB 5.0 out of 5 stars Five Stars nice flat unit , with good sound. Published 11 months ago by Wendy Mitchell 4.0 out of 5 stars Four Stars muy bueno Published 1 year ago by Mario Alejandro Rivetti 5.0 out of 5 stars Five Stars Perfect Published 1 year ago by chevyves 5.0 out of 5 stars Five Stars 100% muy bueno Published 1 year ago by Amazon Customer 5.0 out of 5 stars Five Stars Fantastic Published 1 year ago by Robert

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