Investigating the solutions of $9^x+k(3^x)+2=0$

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












I have worked this one through but still not 100% sure.



  1. the discriminant is $D=(k-2sqrt2)(k+2sqrt2)$.

  2. the quadratic equation gives $3^x=dfrac-kpmsqrtk^2-82$.

  3. as the RHS must be at least $0$ for this equation to have any solutions I did some work using inequalities

I concluded that this equation:



  • cannot have repeated roots when $k=-2sqrt2$


  • has two distinct roots when $k<-2sqrt2$


But not too sure about the details for the case when the equation has two distinct roots. How would you determine this?



Thank you




discriminant




share|cite|improve this question
















  • 1




    yep, you can also check using wolfram...wolframalpha.com/input/?i=9%5Ex%2B8%5E(1%2F2)(3%5Ex)%2B2+%3D+0
    – Andrew Allen
    Aug 28 at 13:52






  • 1




    you must have $3^x > 0$ if you are to finally get an $x$ out of it.
    – Alvin Lepik
    Aug 28 at 13:52














up vote
1
down vote

favorite












I have worked this one through but still not 100% sure.



  1. the discriminant is $D=(k-2sqrt2)(k+2sqrt2)$.

  2. the quadratic equation gives $3^x=dfrac-kpmsqrtk^2-82$.

  3. as the RHS must be at least $0$ for this equation to have any solutions I did some work using inequalities

I concluded that this equation:



  • cannot have repeated roots when $k=-2sqrt2$


  • has two distinct roots when $k<-2sqrt2$


But not too sure about the details for the case when the equation has two distinct roots. How would you determine this?



Thank you




discriminant




share|cite|improve this question
















  • 1




    yep, you can also check using wolfram...wolframalpha.com/input/?i=9%5Ex%2B8%5E(1%2F2)(3%5Ex)%2B2+%3D+0
    – Andrew Allen
    Aug 28 at 13:52






  • 1




    you must have $3^x > 0$ if you are to finally get an $x$ out of it.
    – Alvin Lepik
    Aug 28 at 13:52












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I have worked this one through but still not 100% sure.



  1. the discriminant is $D=(k-2sqrt2)(k+2sqrt2)$.

  2. the quadratic equation gives $3^x=dfrac-kpmsqrtk^2-82$.

  3. as the RHS must be at least $0$ for this equation to have any solutions I did some work using inequalities

I concluded that this equation:



  • cannot have repeated roots when $k=-2sqrt2$


  • has two distinct roots when $k<-2sqrt2$


But not too sure about the details for the case when the equation has two distinct roots. How would you determine this?



Thank you




discriminant




share|cite|improve this question












I have worked this one through but still not 100% sure.



  1. the discriminant is $D=(k-2sqrt2)(k+2sqrt2)$.

  2. the quadratic equation gives $3^x=dfrac-kpmsqrtk^2-82$.

  3. as the RHS must be at least $0$ for this equation to have any solutions I did some work using inequalities

I concluded that this equation:



  • cannot have repeated roots when $k=-2sqrt2$


  • has two distinct roots when $k<-2sqrt2$


But not too sure about the details for the case when the equation has two distinct roots. How would you determine this?



Thank you





discriminant





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 28 at 13:46









Will Kim

1345




1345







  • 1




    yep, you can also check using wolfram...wolframalpha.com/input/?i=9%5Ex%2B8%5E(1%2F2)(3%5Ex)%2B2+%3D+0
    – Andrew Allen
    Aug 28 at 13:52






  • 1




    you must have $3^x > 0$ if you are to finally get an $x$ out of it.
    – Alvin Lepik
    Aug 28 at 13:52












  • 1




    yep, you can also check using wolfram...wolframalpha.com/input/?i=9%5Ex%2B8%5E(1%2F2)(3%5Ex)%2B2+%3D+0
    – Andrew Allen
    Aug 28 at 13:52






  • 1




    you must have $3^x > 0$ if you are to finally get an $x$ out of it.
    – Alvin Lepik
    Aug 28 at 13:52







1




1




yep, you can also check using wolfram...wolframalpha.com/input/?i=9%5Ex%2B8%5E(1%2F2)(3%5Ex)%2B2+%3D+0
– Andrew Allen
Aug 28 at 13:52




yep, you can also check using wolfram...wolframalpha.com/input/?i=9%5Ex%2B8%5E(1%2F2)(3%5Ex)%2B2+%3D+0
– Andrew Allen
Aug 28 at 13:52




1




1




you must have $3^x > 0$ if you are to finally get an $x$ out of it.
– Alvin Lepik
Aug 28 at 13:52




you must have $3^x > 0$ if you are to finally get an $x$ out of it.
– Alvin Lepik
Aug 28 at 13:52










1 Answer
1






active

oldest

votes

















up vote
2
down vote



accepted










To be able to compute the roots of $3^2x + k3^x +2$ with respect to $3^x$ you have immediately the condition $k^2 - 8 geq 0$. So $kleq -2sqrt2$ or $kgeq 2sqrt2$. In case of either equality, you have zero discriminant.



You must also meet the condition
$$-k pm sqrtk^2-8 > 0. $$
This leaves you with $k < -2sqrt2$.



Solving analytically the inequality $-k pm sqrtk^2-8 > 0$ can be prone to mistakes so make sure to double check.



When in doubt, I can always use my rudimentary geogebra skills to draw a picture and get a feeling for what's going on. The green line is $f(x) = -x$ and blue and red lines are $g(x) = sqrtx^2-8$ and $h(x) = -g(x)$ respectively.



enter image description here






share|cite|improve this answer






















  • Thank you. This makes sense now.
    – Will Kim
    Aug 28 at 15:28










Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);













 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2897281%2finvestigating-the-solutions-of-9xk3x2-0%23new-answer', 'question_page');

);

Post as a guest

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote



accepted










To be able to compute the roots of $3^2x + k3^x +2$ with respect to $3^x$ you have immediately the condition $k^2 - 8 geq 0$. So $kleq -2sqrt2$ or $kgeq 2sqrt2$. In case of either equality, you have zero discriminant.



You must also meet the condition
$$-k pm sqrtk^2-8 > 0. $$
This leaves you with $k < -2sqrt2$.



Solving analytically the inequality $-k pm sqrtk^2-8 > 0$ can be prone to mistakes so make sure to double check.



When in doubt, I can always use my rudimentary geogebra skills to draw a picture and get a feeling for what's going on. The green line is $f(x) = -x$ and blue and red lines are $g(x) = sqrtx^2-8$ and $h(x) = -g(x)$ respectively.



enter image description here






share|cite|improve this answer






















  • Thank you. This makes sense now.
    – Will Kim
    Aug 28 at 15:28














up vote
2
down vote



accepted










To be able to compute the roots of $3^2x + k3^x +2$ with respect to $3^x$ you have immediately the condition $k^2 - 8 geq 0$. So $kleq -2sqrt2$ or $kgeq 2sqrt2$. In case of either equality, you have zero discriminant.



You must also meet the condition
$$-k pm sqrtk^2-8 > 0. $$
This leaves you with $k < -2sqrt2$.



Solving analytically the inequality $-k pm sqrtk^2-8 > 0$ can be prone to mistakes so make sure to double check.



When in doubt, I can always use my rudimentary geogebra skills to draw a picture and get a feeling for what's going on. The green line is $f(x) = -x$ and blue and red lines are $g(x) = sqrtx^2-8$ and $h(x) = -g(x)$ respectively.



enter image description here






share|cite|improve this answer






















  • Thank you. This makes sense now.
    – Will Kim
    Aug 28 at 15:28












up vote
2
down vote



accepted







up vote
2
down vote



accepted






To be able to compute the roots of $3^2x + k3^x +2$ with respect to $3^x$ you have immediately the condition $k^2 - 8 geq 0$. So $kleq -2sqrt2$ or $kgeq 2sqrt2$. In case of either equality, you have zero discriminant.



You must also meet the condition
$$-k pm sqrtk^2-8 > 0. $$
This leaves you with $k < -2sqrt2$.



Solving analytically the inequality $-k pm sqrtk^2-8 > 0$ can be prone to mistakes so make sure to double check.



When in doubt, I can always use my rudimentary geogebra skills to draw a picture and get a feeling for what's going on. The green line is $f(x) = -x$ and blue and red lines are $g(x) = sqrtx^2-8$ and $h(x) = -g(x)$ respectively.



enter image description here






share|cite|improve this answer














To be able to compute the roots of $3^2x + k3^x +2$ with respect to $3^x$ you have immediately the condition $k^2 - 8 geq 0$. So $kleq -2sqrt2$ or $kgeq 2sqrt2$. In case of either equality, you have zero discriminant.



You must also meet the condition
$$-k pm sqrtk^2-8 > 0. $$
This leaves you with $k < -2sqrt2$.



Solving analytically the inequality $-k pm sqrtk^2-8 > 0$ can be prone to mistakes so make sure to double check.



When in doubt, I can always use my rudimentary geogebra skills to draw a picture and get a feeling for what's going on. The green line is $f(x) = -x$ and blue and red lines are $g(x) = sqrtx^2-8$ and $h(x) = -g(x)$ respectively.



enter image description here







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 28 at 14:41

























answered Aug 28 at 13:59









Alvin Lepik

2,528921




2,528921











  • Thank you. This makes sense now.
    – Will Kim
    Aug 28 at 15:28
















  • Thank you. This makes sense now.
    – Will Kim
    Aug 28 at 15:28















Thank you. This makes sense now.
– Will Kim
Aug 28 at 15:28




Thank you. This makes sense now.
– Will Kim
Aug 28 at 15:28

















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2897281%2finvestigating-the-solutions-of-9xk3x2-0%23new-answer', 'question_page');

);

Post as a guest
































































-

這個網誌中的熱門文章

How to combine Bézier curves to a surface?

圖片
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier curve. Additionally it marks one side of the curve as inside t
閱讀完整內容

Why am i infinitely getting the same tweet with the Twitter Search API?

圖片
up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia
閱讀完整內容

Is there any way to eliminate the singular point to solve this integral by hand or by approximations?

圖片
Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite 1 $$int_0^1 (1+(x^-p-1)^1-p)^frac1pdx, pin mathbbR, pge 1$$ I find that $x=0,1$ are singular points(discontinuous points of this integral), is there a way to eliminate those singular points to make sure that this integral can be calculated by hand or by Taylor approximations. In either cases, what are those steps of calculating the integral? calculus real-analysis analysis approximation share | cite | improve this question edited Sep 11 at 5:07 asked Sep 11 at 4:48 Mclalalala 126 8 1 What is $p$? Please give more information to make it complete. – xbh Sep 11 at 5:02 @xbh already fixed it, thanks for reminding. – Mclalalala Sep 11 at 5:10 Seems that $0$ is not singular here… [I might compute wrong] – xbh Sep 11 at 5:30 @xbh can you show me how to get the close formula for the integral above? – Mclalalala Sep 11 at 5:33
閱讀完整內容