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(sqrt(x+4)+sqrt(x+4))/2=x+sqrt(x^(2)-16)-6 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
  _______     _______          _________    
\/ x + 4  + \/ x + 4          /  2          
--------------------- = x + \/  x  - 16  - 6
          2                                 
$$\frac{\sqrt{x + 4} + \sqrt{x + 4}}{2} = \left(x + \sqrt{x^{2} - 16}\right) - 6$$
Gráfica
Suma y producto de raíces [src]
suma
            /                     1                     \     /       /                     1                     \                                                                      \                                                                      
     3067*re|-------------------------------------------|     |3067*im|-------------------------------------------|               /    /       _____\\               /    /       _____\\|               /    /       _____\\               /    /       _____\\
            |                     ______________________|     |       |                     ______________________|               |    |1600*\/ 669 ||               |    |1600*\/ 669 |||               |    |1600*\/ 669 ||               |    |1600*\/ 669 ||
            |/          ___\     /                _____ |     |       |/          ___\     /                _____ |               |atan|------------||               |atan|------------|||               |atan|------------||               |atan|------------||
            ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     |       ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     ______    |    \   291267   /|     ______    |    \   291267   /||     ______    |    \   291267   /|     ______    |    \   291267   /|
            ||- - - -------|*3 /   ----- + ------------ |     |       ||- - - -------|*3 /   ----- + ------------ |   \/ 3067 *cos|------------------|   \/ 9201 *sin|------------------||   \/ 9201 *cos|------------------|   \/ 3067 *sin|------------------|
51          \\  2      2   / \/      64         9       /     |       \\  2      2   / \/      64         9       /               \        3         /               \        3         /|               \        3         /               \        3         /
-- + ---------------------------------------------------- + I*|---------------------------------------------------- - -------------------------------- - --------------------------------| - -------------------------------- + --------------------------------
4                             48                              \                         48                                           8                                  24               /                  24                                 8                
$$- \frac{\sqrt{9201} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48} + \frac{\sqrt{3067} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} + \frac{51}{4} + i \left(- \frac{\sqrt{3067} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} - \frac{\sqrt{9201} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48}\right)$$
=
            /                     1                     \     /       /                     1                     \                                                                      \                                                                      
     3067*re|-------------------------------------------|     |3067*im|-------------------------------------------|               /    /       _____\\               /    /       _____\\|               /    /       _____\\               /    /       _____\\
            |                     ______________________|     |       |                     ______________________|               |    |1600*\/ 669 ||               |    |1600*\/ 669 |||               |    |1600*\/ 669 ||               |    |1600*\/ 669 ||
            |/          ___\     /                _____ |     |       |/          ___\     /                _____ |               |atan|------------||               |atan|------------|||               |atan|------------||               |atan|------------||
            ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     |       ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     ______    |    \   291267   /|     ______    |    \   291267   /||     ______    |    \   291267   /|     ______    |    \   291267   /|
            ||- - - -------|*3 /   ----- + ------------ |     |       ||- - - -------|*3 /   ----- + ------------ |   \/ 3067 *cos|------------------|   \/ 9201 *sin|------------------||   \/ 9201 *cos|------------------|   \/ 3067 *sin|------------------|
51          \\  2      2   / \/      64         9       /     |       \\  2      2   / \/      64         9       /               \        3         /               \        3         /|               \        3         /               \        3         /
-- + ---------------------------------------------------- + I*|---------------------------------------------------- - -------------------------------- - --------------------------------| - -------------------------------- + --------------------------------
4                             48                              \                         48                                           8                                  24               /                  24                                 8                
$$- \frac{\sqrt{9201} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48} + \frac{\sqrt{3067} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} + \frac{51}{4} + i \left(- \frac{\sqrt{3067} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} - \frac{\sqrt{9201} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48}\right)$$
producto
            /                     1                     \     /       /                     1                     \                                                                      \                                                                      
     3067*re|-------------------------------------------|     |3067*im|-------------------------------------------|               /    /       _____\\               /    /       _____\\|               /    /       _____\\               /    /       _____\\
            |                     ______________________|     |       |                     ______________________|               |    |1600*\/ 669 ||               |    |1600*\/ 669 |||               |    |1600*\/ 669 ||               |    |1600*\/ 669 ||
            |/          ___\     /                _____ |     |       |/          ___\     /                _____ |               |atan|------------||               |atan|------------|||               |atan|------------||               |atan|------------||
            ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     |       ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     ______    |    \   291267   /|     ______    |    \   291267   /||     ______    |    \   291267   /|     ______    |    \   291267   /|
            ||- - - -------|*3 /   ----- + ------------ |     |       ||- - - -------|*3 /   ----- + ------------ |   \/ 3067 *cos|------------------|   \/ 9201 *sin|------------------||   \/ 9201 *cos|------------------|   \/ 3067 *sin|------------------|
51          \\  2      2   / \/      64         9       /     |       \\  2      2   / \/      64         9       /               \        3         /               \        3         /|               \        3         /               \        3         /
-- + ---------------------------------------------------- + I*|---------------------------------------------------- - -------------------------------- - --------------------------------| - -------------------------------- + --------------------------------
4                             48                              \                         48                                           8                                  24               /                  24                                 8                
$$- \frac{\sqrt{9201} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48} + \frac{\sqrt{3067} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} + \frac{51}{4} + i \left(- \frac{\sqrt{3067} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} - \frac{\sqrt{9201} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48}\right)$$
=
                 /      /       _____\     \
                 |      |1600*\/ 669 |     |
                 |  atan|------------|     |
       ______    |      \   291267   /   pi|
     \/ 9201 *sin|- ------------------ + --|
51               \          3            6 /
-- - ---------------------------------------
4                       6                   
$$- \frac{\sqrt{9201} \sin{\left(- \frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} + \frac{\pi}{6} \right)}}{6} + \frac{51}{4}$$
51/4 - sqrt(9201)*sin(-atan(1600*sqrt(669)/291267)/3 + pi/6)/6
Respuesta rápida [src]
                 /                     1                     \     /       /                     1                     \                                                                      \                                                                      
          3067*re|-------------------------------------------|     |3067*im|-------------------------------------------|               /    /       _____\\               /    /       _____\\|               /    /       _____\\               /    /       _____\\
                 |                     ______________________|     |       |                     ______________________|               |    |1600*\/ 669 ||               |    |1600*\/ 669 |||               |    |1600*\/ 669 ||               |    |1600*\/ 669 ||
                 |/          ___\     /                _____ |     |       |/          ___\     /                _____ |               |atan|------------||               |atan|------------|||               |atan|------------||               |atan|------------||
                 ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     |       ||  1   I*\/ 3 |    /  32363   25*I*\/ 669  |     ______    |    \   291267   /|     ______    |    \   291267   /||     ______    |    \   291267   /|     ______    |    \   291267   /|
                 ||- - - -------|*3 /   ----- + ------------ |     |       ||- - - -------|*3 /   ----- + ------------ |   \/ 3067 *cos|------------------|   \/ 9201 *sin|------------------||   \/ 9201 *cos|------------------|   \/ 3067 *sin|------------------|
     51          \\  2      2   / \/      64         9       /     |       \\  2      2   / \/      64         9       /               \        3         /               \        3         /|               \        3         /               \        3         /
x1 = -- + ---------------------------------------------------- + I*|---------------------------------------------------- - -------------------------------- - --------------------------------| - -------------------------------- + --------------------------------
     4                             48                              \                         48                                           8                                  24               /                  24                                 8                
$$x_{1} = - \frac{\sqrt{9201} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48} + \frac{\sqrt{3067} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} + \frac{51}{4} + i \left(- \frac{\sqrt{3067} \cos{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{8} - \frac{\sqrt{9201} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1600 \sqrt{669}}{291267} \right)}}{3} \right)}}{24} + \frac{3067 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{32363}{64} + \frac{25 \sqrt{669} i}{9}}}\right)}}{48}\right)$$
x1 = -sqrt(9201)*cos(atan(1600*sqrt(669)/291267)/3)/24 + 3067*re(1/((-1/2 - sqrt(3)*i/2)*(32363/64 + 25*sqrt(669)*i/9)^(1/3)))/48 + sqrt(3067)*sin(atan(1600*sqrt(669)/291267)/3)/8 + 51/4 + i*(-sqrt(3067)*cos(atan(1600*sqrt(669)/291267)/3)/8 - sqrt(9201)*sin(atan(1600*sqrt(669)/291267)/3)/24 + 3067*im(1/((-1/2 - sqrt(3)*i/2)*(32363/64 + 25*sqrt(669)*i/9)^(1/3)))/48)
Respuesta numérica [src]
x1 = 5.41647598981574
x1 = 5.41647598981574