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3sqrt(x^2(x-1))=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

     ____________    
    /  2             
3*\/  x *(x - 1)  = 0

$$3 \sqrt{x^{2} \left(x - 1\right)} = 0$$

Simplificar

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Respuesta rápida [src]

x1 = 0

$$x_{1} = 0$$

x2 = 1

$$x_{2} = 1$$

         3 ____       2/3
     1   \/ -1    (-1)   
x3 = - - ------ + -------
     3     3         3

$$x_{3} = \frac{1}{3} - \frac{\sqrt[3]{-1}}{3} + \frac{\left(-1\right)^{\frac{2}{3}}}{3}$$

           /       2/3   \       /       2/3   \
           |   (-1)      |       |   (-1)      |
         re|-------------|   I*im|-------------|
           |          ___|       |          ___|
           |  1   I*\/ 3 |       |  1   I*\/ 3 |
           |- - + -------|       |- - + -------|
     2     \  2      2   /       \  2      2   /
x4 = - + ----------------- + -------------------
     3           3                    3

$$x_{4} = \frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{2}{3} + \frac{i \operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3}$$

           /       2/3   \     /  /       2/3   \        \
           |   (-1)      |     |  |   (-1)      |        |
         re|-------------|     |im|-------------|        |
           |          ___|     |  |          ___|        |
           |  1   I*\/ 3 |     |  |  1   I*\/ 3 |        |
           |- - - -------|     |  |- - - -------|     ___|
     1     \  2      2   /     |  \  2      2   /   \/ 3 |
x5 = - + ----------------- + I*|----------------- + -----|
     6           3             \        3             6  /

$$x_{5} = \frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{1}{6} + i \left(\frac{\operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{\sqrt{3}}{6}\right)$$

x5 = re((-1)^(2/3)/(-1/2 - sqrt(3)*i/2))/3 + 1/6 + i*(im((-1)^(2/3)/(-1/2 - sqrt(3)*i/2))/3 + sqrt(3)/6)

Suma y producto de raíces [src]

suma

                                 /       2/3   \       /       2/3   \         /       2/3   \     /  /       2/3   \        \
                                 |   (-1)      |       |   (-1)      |         |   (-1)      |     |  |   (-1)      |        |
                               re|-------------|   I*im|-------------|       re|-------------|     |im|-------------|        |
                                 |          ___|       |          ___|         |          ___|     |  |          ___|        |
                                 |  1   I*\/ 3 |       |  1   I*\/ 3 |         |  1   I*\/ 3 |     |  |  1   I*\/ 3 |        |
        3 ____       2/3         |- - + -------|       |- - + -------|         |- - - -------|     |  |- - - -------|     ___|
    1   \/ -1    (-1)      2     \  2      2   /       \  2      2   /   1     \  2      2   /     |  \  2      2   /   \/ 3 |
1 + - - ------ + ------- + - + ----------------- + ------------------- + - + ----------------- + I*|----------------- + -----|
    3     3         3      3           3                    3            6           3             \        3             6  /

$$\left(\left(\frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{2}{3} + \frac{i \operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3}\right) + \left(1 + \left(\frac{1}{3} - \frac{\sqrt[3]{-1}}{3} + \frac{\left(-1\right)^{\frac{2}{3}}}{3}\right)\right)\right) + \left(\frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{1}{6} + i \left(\frac{\operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{\sqrt{3}}{6}\right)\right)$$

                          /       2/3   \     /       2/3   \     /  /       2/3   \        \       /       2/3   \
                          |   (-1)      |     |   (-1)      |     |  |   (-1)      |        |       |   (-1)      |
                        re|-------------|   re|-------------|     |im|-------------|        |   I*im|-------------|
                          |          ___|     |          ___|     |  |          ___|        |       |          ___|
                          |  1   I*\/ 3 |     |  1   I*\/ 3 |     |  |  1   I*\/ 3 |        |       |  1   I*\/ 3 |
     3 ____       2/3     |- - + -------|     |- - - -------|     |  |- - - -------|     ___|       |- - + -------|
13   \/ -1    (-1)        \  2      2   /     \  2      2   /     |  \  2      2   /   \/ 3 |       \  2      2   /
-- - ------ + ------- + ----------------- + ----------------- + I*|----------------- + -----| + -------------------
6      3         3              3                   3             \        3             6  /            3

$$\frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{13}{6} - \frac{\sqrt[3]{-1}}{3} + \frac{i \operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3} + i \left(\frac{\operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{\sqrt{3}}{6}\right) + \frac{\left(-1\right)^{\frac{2}{3}}}{3}$$

producto

                         /      /       2/3   \       /       2/3   \\ /      /       2/3   \     /  /       2/3   \        \\
                         |      |   (-1)      |       |   (-1)      || |      |   (-1)      |     |  |   (-1)      |        ||
                         |    re|-------------|   I*im|-------------|| |    re|-------------|     |im|-------------|        ||
                         |      |          ___|       |          ___|| |      |          ___|     |  |          ___|        ||
                         |      |  1   I*\/ 3 |       |  1   I*\/ 3 || |      |  1   I*\/ 3 |     |  |  1   I*\/ 3 |        ||
  /    3 ____       2/3\ |      |- - + -------|       |- - + -------|| |      |- - - -------|     |  |- - - -------|     ___||
  |1   \/ -1    (-1)   | |2     \  2      2   /       \  2      2   /| |1     \  2      2   /     |  \  2      2   /   \/ 3 ||
0*|- - ------ + -------|*|- + ----------------- + -------------------|*|- + ----------------- + I*|----------------- + -----||
  \3     3         3   / \3           3                    3         / \6           3             \        3             6  //

$$0 \left(\frac{1}{3} - \frac{\sqrt[3]{-1}}{3} + \frac{\left(-1\right)^{\frac{2}{3}}}{3}\right) \left(\frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{2}{3} + \frac{i \operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)}}{3}\right) \left(\frac{\operatorname{re}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{1}{6} + i \left(\frac{\operatorname{im}{\left(\frac{\left(-1\right)^{\frac{2}{3}}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}}\right)}}{3} + \frac{\sqrt{3}}{6}\right)\right)$$

$$0$$

Respuesta numérica [src]

x1 = -0.e-126 + 0.e-125*i

x2 = 1.0 - 6.61744490042422e-24*i

x3 = -0.e-126 + 0.e-128*i

x3 = -0.e-126 + 0.e-128*i

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Expresiones con funciones

3*sqrt(x^2*(x-1))=0 la ecuación

Solución numérica:

Solución

3sqrt(x^2(x-1))=0 la ecuación