What are the possible solutions of the given equation?When do we get extraneous roots?Algebraic manipulation with square rootsFind all real numbers such that $sqrtx-frac1x + sqrt1 - frac1x = x$Transposing an equation: x = F/k + sqrt(F/c) to get F as the subjectFinding all real roots of the equation $(x+1) sqrtx+2 + (x+6)sqrtx+7 = x^2+7x+12$Justify the algorithm used to create a polynomial whose roots are squares of the roots of the given polynomial (over $mathbb C$)Integral solutions to the equation $left(frac1nright)^-1/2=sqrta+sqrt15-sqrta-sqrt15.$What rule governs $x^4=10,000$ having complex solutions?Why do I keep getting this incorrect solution when trying to find all the real solutions for $sqrt2x-3 +x=3$.Radical equation - can I square both sides with more than 1 radical on one side?

Russian cases: A few examples, I'm really confused

Informing my boss about remarks from a nasty colleague

Do I need life insurance if I can cover my own funeral costs?

How to make healing in an exploration game interesting

How to simplify this time periods definition interface?

Welcoming 2019 Pi day: How to draw the letter π?

Is it normal that my co-workers at a fitness company criticize my food choices?

Sword in the Stone story where the sword was held in place by electromagnets

Why must traveling waves have the same amplitude to form a standing wave?

Does splitting a potentially monolithic application into several smaller ones help prevent bugs?

Could the Saturn V actually have launched astronauts around Venus?

PTIJ: Who should pay for Uber rides: the child or the parent?

Where is the 1/8 CR apprentice in Volo's Guide to Monsters?

Humanity loses the vast majority of its technology, information, and population in the year 2122. How long does it take to rebuild itself?

PTIJ: is Mi Yodeya found in the Torah codes?

Identifying the interval from A♭ to D♯

2D counterpart of std::array in C++17

Official degrees of earth’s rotation per day

Rules about breaking the rules. How do I do it well?

How to answer questions about my characters?

Why does Deadpool say "You're welcome, Canada," after shooting Ryan Reynolds in the end credits?

Bastion server: use TCP forwarding VS placing private key on server

Possible Leak In Concrete

Why did it take so long to abandon sail after steamships were demonstrated?



What are the possible solutions of the given equation?


When do we get extraneous roots?Algebraic manipulation with square rootsFind all real numbers such that $sqrtx-frac1x + sqrt1 - frac1x = x$Transposing an equation: x = F/k + sqrt(F/c) to get F as the subjectFinding all real roots of the equation $(x+1) sqrtx+2 + (x+6)sqrtx+7 = x^2+7x+12$Justify the algorithm used to create a polynomial whose roots are squares of the roots of the given polynomial (over $mathbb C$)Integral solutions to the equation $left(frac1nright)^-1/2=sqrta+sqrt15-sqrta-sqrt15.$What rule governs $x^4=10,000$ having complex solutions?Why do I keep getting this incorrect solution when trying to find all the real solutions for $sqrt2x-3 +x=3$.Radical equation - can I square both sides with more than 1 radical on one side?













2












$begingroup$


I encountered a question in an exam in which we had:




Find all possible solutions of the equation $$x+y+ 1over x+1over y+4=2 (sqrt 2x+1+sqrt 2y+1) $$ where $x $ and $y$ are real numbers.




I tried squaring both sides to eliminate the square roots but the number of terms became too many, making the problem very difficult to handle. I am not really able to understand how to find an easier approach or handle the terms efficiently. Would someone please help me to solve this question?










share|cite|improve this question











$endgroup$
















    2












    $begingroup$


    I encountered a question in an exam in which we had:




    Find all possible solutions of the equation $$x+y+ 1over x+1over y+4=2 (sqrt 2x+1+sqrt 2y+1) $$ where $x $ and $y$ are real numbers.




    I tried squaring both sides to eliminate the square roots but the number of terms became too many, making the problem very difficult to handle. I am not really able to understand how to find an easier approach or handle the terms efficiently. Would someone please help me to solve this question?










    share|cite|improve this question











    $endgroup$














      2












      2








      2





      $begingroup$


      I encountered a question in an exam in which we had:




      Find all possible solutions of the equation $$x+y+ 1over x+1over y+4=2 (sqrt 2x+1+sqrt 2y+1) $$ where $x $ and $y$ are real numbers.




      I tried squaring both sides to eliminate the square roots but the number of terms became too many, making the problem very difficult to handle. I am not really able to understand how to find an easier approach or handle the terms efficiently. Would someone please help me to solve this question?










      share|cite|improve this question











      $endgroup$




      I encountered a question in an exam in which we had:




      Find all possible solutions of the equation $$x+y+ 1over x+1over y+4=2 (sqrt 2x+1+sqrt 2y+1) $$ where $x $ and $y$ are real numbers.




      I tried squaring both sides to eliminate the square roots but the number of terms became too many, making the problem very difficult to handle. I am not really able to understand how to find an easier approach or handle the terms efficiently. Would someone please help me to solve this question?







      algebra-precalculus






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago









      Thomas Andrews

      130k12147298




      130k12147298










      asked 4 hours ago









      Shashwat1337Shashwat1337

      889




      889




















          2 Answers
          2






          active

          oldest

          votes


















          7












          $begingroup$

          $$x+y+ 1over x+1over y+4 = x+y+2x+1over x+2y+1over y $$



          By Am-Gm we have $$ x+2x+1over xgeq 2sqrtx2x+1over x = 2sqrt2x+1$$ and the same for $y$, so we have



          $$x+y+ 1over x+1over y+4 geq 2sqrt2x+1+2sqrt2y+1$$



          Since we have equality is achieved when $x=2x+1over x$ (and the same for $y$) we have $x=y=1+sqrt2$






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            Truly amazing!!! [+1]
            $endgroup$
            – Dr. Mathva
            4 hours ago










          • $begingroup$
            @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
            $endgroup$
            – Michael Rozenberg
            4 hours ago











          • $begingroup$
            AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
            $endgroup$
            – Anurag A
            4 hours ago



















          2












          $begingroup$

          It's $$sum_cycleft(x+frac1x+2-2sqrt2x+1right)=0$$ or
          $$sum_cycfracx^2-2xsqrt2x+1+2x+1x=0$$ or
          $$sum_cycfrac(x-sqrt2x+1)^2x=0,$$
          which for $xy<0$ gives infinitely many solutions.



          But, for $xy>0$ we obtain:
          $$x=sqrt2x+1$$ and $$y=sqrt2y+1,$$ which gives
          $$x=y=1+sqrt2.$$






          share|cite|improve this answer









          $endgroup$












            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',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader:
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            ,
            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%2f3148348%2fwhat-are-the-possible-solutions-of-the-given-equation%23new-answer', 'question_page');

            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            7












            $begingroup$

            $$x+y+ 1over x+1over y+4 = x+y+2x+1over x+2y+1over y $$



            By Am-Gm we have $$ x+2x+1over xgeq 2sqrtx2x+1over x = 2sqrt2x+1$$ and the same for $y$, so we have



            $$x+y+ 1over x+1over y+4 geq 2sqrt2x+1+2sqrt2y+1$$



            Since we have equality is achieved when $x=2x+1over x$ (and the same for $y$) we have $x=y=1+sqrt2$






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Truly amazing!!! [+1]
              $endgroup$
              – Dr. Mathva
              4 hours ago










            • $begingroup$
              @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
              $endgroup$
              – Michael Rozenberg
              4 hours ago











            • $begingroup$
              AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
              $endgroup$
              – Anurag A
              4 hours ago
















            7












            $begingroup$

            $$x+y+ 1over x+1over y+4 = x+y+2x+1over x+2y+1over y $$



            By Am-Gm we have $$ x+2x+1over xgeq 2sqrtx2x+1over x = 2sqrt2x+1$$ and the same for $y$, so we have



            $$x+y+ 1over x+1over y+4 geq 2sqrt2x+1+2sqrt2y+1$$



            Since we have equality is achieved when $x=2x+1over x$ (and the same for $y$) we have $x=y=1+sqrt2$






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Truly amazing!!! [+1]
              $endgroup$
              – Dr. Mathva
              4 hours ago










            • $begingroup$
              @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
              $endgroup$
              – Michael Rozenberg
              4 hours ago











            • $begingroup$
              AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
              $endgroup$
              – Anurag A
              4 hours ago














            7












            7








            7





            $begingroup$

            $$x+y+ 1over x+1over y+4 = x+y+2x+1over x+2y+1over y $$



            By Am-Gm we have $$ x+2x+1over xgeq 2sqrtx2x+1over x = 2sqrt2x+1$$ and the same for $y$, so we have



            $$x+y+ 1over x+1over y+4 geq 2sqrt2x+1+2sqrt2y+1$$



            Since we have equality is achieved when $x=2x+1over x$ (and the same for $y$) we have $x=y=1+sqrt2$






            share|cite|improve this answer











            $endgroup$



            $$x+y+ 1over x+1over y+4 = x+y+2x+1over x+2y+1over y $$



            By Am-Gm we have $$ x+2x+1over xgeq 2sqrtx2x+1over x = 2sqrt2x+1$$ and the same for $y$, so we have



            $$x+y+ 1over x+1over y+4 geq 2sqrt2x+1+2sqrt2y+1$$



            Since we have equality is achieved when $x=2x+1over x$ (and the same for $y$) we have $x=y=1+sqrt2$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 4 hours ago

























            answered 4 hours ago









            Maria MazurMaria Mazur

            46.9k1260120




            46.9k1260120











            • $begingroup$
              Truly amazing!!! [+1]
              $endgroup$
              – Dr. Mathva
              4 hours ago










            • $begingroup$
              @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
              $endgroup$
              – Michael Rozenberg
              4 hours ago











            • $begingroup$
              AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
              $endgroup$
              – Anurag A
              4 hours ago

















            • $begingroup$
              Truly amazing!!! [+1]
              $endgroup$
              – Dr. Mathva
              4 hours ago










            • $begingroup$
              @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
              $endgroup$
              – Michael Rozenberg
              4 hours ago











            • $begingroup$
              AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
              $endgroup$
              – Anurag A
              4 hours ago
















            $begingroup$
            Truly amazing!!! [+1]
            $endgroup$
            – Dr. Mathva
            4 hours ago




            $begingroup$
            Truly amazing!!! [+1]
            $endgroup$
            – Dr. Mathva
            4 hours ago












            $begingroup$
            @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
            $endgroup$
            – Michael Rozenberg
            4 hours ago





            $begingroup$
            @Maria Mazur Your solution is total wrong. What happens for $xy<0?$ If you'll see down-voting it's not mine.
            $endgroup$
            – Michael Rozenberg
            4 hours ago













            $begingroup$
            AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
            $endgroup$
            – Anurag A
            4 hours ago





            $begingroup$
            AM-GM requires terms to be positive. So your solution doesn't account for the case when terms are negative.
            $endgroup$
            – Anurag A
            4 hours ago












            2












            $begingroup$

            It's $$sum_cycleft(x+frac1x+2-2sqrt2x+1right)=0$$ or
            $$sum_cycfracx^2-2xsqrt2x+1+2x+1x=0$$ or
            $$sum_cycfrac(x-sqrt2x+1)^2x=0,$$
            which for $xy<0$ gives infinitely many solutions.



            But, for $xy>0$ we obtain:
            $$x=sqrt2x+1$$ and $$y=sqrt2y+1,$$ which gives
            $$x=y=1+sqrt2.$$






            share|cite|improve this answer









            $endgroup$

















              2












              $begingroup$

              It's $$sum_cycleft(x+frac1x+2-2sqrt2x+1right)=0$$ or
              $$sum_cycfracx^2-2xsqrt2x+1+2x+1x=0$$ or
              $$sum_cycfrac(x-sqrt2x+1)^2x=0,$$
              which for $xy<0$ gives infinitely many solutions.



              But, for $xy>0$ we obtain:
              $$x=sqrt2x+1$$ and $$y=sqrt2y+1,$$ which gives
              $$x=y=1+sqrt2.$$






              share|cite|improve this answer









              $endgroup$















                2












                2








                2





                $begingroup$

                It's $$sum_cycleft(x+frac1x+2-2sqrt2x+1right)=0$$ or
                $$sum_cycfracx^2-2xsqrt2x+1+2x+1x=0$$ or
                $$sum_cycfrac(x-sqrt2x+1)^2x=0,$$
                which for $xy<0$ gives infinitely many solutions.



                But, for $xy>0$ we obtain:
                $$x=sqrt2x+1$$ and $$y=sqrt2y+1,$$ which gives
                $$x=y=1+sqrt2.$$






                share|cite|improve this answer









                $endgroup$



                It's $$sum_cycleft(x+frac1x+2-2sqrt2x+1right)=0$$ or
                $$sum_cycfracx^2-2xsqrt2x+1+2x+1x=0$$ or
                $$sum_cycfrac(x-sqrt2x+1)^2x=0,$$
                which for $xy<0$ gives infinitely many solutions.



                But, for $xy>0$ we obtain:
                $$x=sqrt2x+1$$ and $$y=sqrt2y+1,$$ which gives
                $$x=y=1+sqrt2.$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 4 hours ago









                Michael RozenbergMichael Rozenberg

                108k1895200




                108k1895200



























                    draft saved

                    draft discarded
















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid


                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.

                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3148348%2fwhat-are-the-possible-solutions-of-the-given-equation%23new-answer', 'question_page');

                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Can not update quote_id field of “quote_item” table magento 2Magento 2.1 - We can't remove the item. (Shopping Cart doesnt allow us to remove items before becomes empty)Add value for custom quote item attribute using REST apiREST API endpoint v1/carts/cartId/items always returns error messageCorrect way to save entries to databaseHow to remove all associated quote objects of a customer completelyMagento 2 - Save value from custom input field to quote_itemGet quote_item data using quote id and product id filter in Magento 2How to set additional data to quote_item table from controller in Magento 2?What is the purpose of additional_data column in quote_item table in magento2Set Custom Price to Quote item magento2 from controller

                    How to solve knockout JS error in Magento 2 Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?(Magento2) knockout.js:3012 Uncaught ReferenceError: Unable to process bindingUnable to process binding Knockout.js magento 2Cannot read property `scopeLabel` of undefined on Product Detail PageCan't get Customer Data on frontend in Magento 2Magento2 Order Summary - unable to process bindingKO templates are not loading in Magento 2.1 applicationgetting knockout js error magento 2Product grid not load -— Unable to process binding Knockout.js magento 2Product form not loaded in magento2Uncaught ReferenceError: Unable to process binding “if: function()return (isShowLegend()) ” magento 2

                    Nissan Patrol Зміст Перше покоління — 4W60 (1951-1960) | Друге покоління — 60 series (1960-1980) | Третє покоління (1980–2002) | Четверте покоління — Y60 (1987–1998) | П'яте покоління — Y61 (1997–2013) | Шосте покоління — Y62 (2010- ) | Посилання | Зноски | Навігаційне менюОфіційний український сайтТест-драйв Nissan Patrol 2010 7-го поколінняNissan PatrolКак мы тестировали Nissan Patrol 2016рвиправивши або дописавши її