Sr Examen
18x^3-4x+18x^2=6,5 la ecuación
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Solución
Ha introducido
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3 2 18*x - 4*x + 18*x = 13/2
$$18 x^{2} + \left(18 x^{3} - 4 x\right) = \frac{13}{2}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$18 x^{2} + \left(18 x^{3} - 4 x\right) = \frac{13}{2}$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + x^{2} - \frac{2 x}{9} - \frac{13}{36} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{9}$$
$$v = \frac{d}{a}$$
$$v = - \frac{13}{36}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{2}{9}$$
$$x_{1} x_{2} x_{3} = - \frac{13}{36}$$
$$18 x^{2} + \left(18 x^{3} - 4 x\right) = \frac{13}{2}$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + x^{2} - \frac{2 x}{9} - \frac{13}{36} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{9}$$
$$v = \frac{d}{a}$$
$$v = - \frac{13}{36}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{2}{9}$$
$$x_{1} x_{2} x_{3} = - \frac{13}{36}$$
Respuesta rápida
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_________________ / _________________ \
/ _______ | / _______ |
/ 23 \/ 18849 | ___ / 23 \/ 18849 |
3 / --- + --------- | \/ 3 *3 / --- + --------- ___ |
1 5 \/ 216 1944 | \/ 216 1944 5*\/ 3 |
x1 = - - - ------------------------- - ---------------------- + I*|- ---------------------------- + -------------------------|
3 _________________ 2 | 2 _________________|
/ _______ | / _______ |
/ 23 \/ 18849 | / 23 \/ 18849 |
54*3 / --- + --------- | 54*3 / --- + --------- |
\/ 216 1944 \ \/ 216 1944 /
$$x_{1} = - \frac{1}{3} - \frac{\sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} - \frac{5}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} + \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}\right)$$
_________________ / _________________ \
/ _______ | / _______ |
/ 23 \/ 18849 | ___ / 23 \/ 18849 |
3 / --- + --------- |\/ 3 *3 / --- + --------- ___ |
1 5 \/ 216 1944 | \/ 216 1944 5*\/ 3 |
x2 = - - - ------------------------- - ---------------------- + I*|---------------------------- - -------------------------|
3 _________________ 2 | 2 _________________|
/ _______ | / _______ |
/ 23 \/ 18849 | / 23 \/ 18849 |
54*3 / --- + --------- | 54*3 / --- + --------- |
\/ 216 1944 \ \/ 216 1944 /
$$x_{2} = - \frac{1}{3} - \frac{\sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} - \frac{5}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + i \left(- \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2}\right)$$
_________________
/ _______
1 / 23 \/ 18849 5
x3 = - - + 3 / --- + --------- + -------------------------
3 \/ 216 1944 _________________
/ _______
/ 23 \/ 18849
27*3 / --- + ---------
\/ 216 1944
$$x_{3} = - \frac{1}{3} + \frac{5}{27 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}$$
x3 = -1/3 + 5/(27*(sqrt(18849)/1944 + 23/216)^(1/3)) + (sqrt(18849)/1944 + 23/216)^(1/3)
Suma y producto de raíces
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suma
_________________ / _________________ \ _________________ / _________________ \
/ _______ | / _______ | / _______ | / _______ |
/ 23 \/ 18849 | ___ / 23 \/ 18849 | / 23 \/ 18849 | ___ / 23 \/ 18849 | _________________
3 / --- + --------- | \/ 3 *3 / --- + --------- ___ | 3 / --- + --------- |\/ 3 *3 / --- + --------- ___ | / _______
1 5 \/ 216 1944 | \/ 216 1944 5*\/ 3 | 1 5 \/ 216 1944 | \/ 216 1944 5*\/ 3 | 1 / 23 \/ 18849 5
- - - ------------------------- - ---------------------- + I*|- ---------------------------- + -------------------------| + - - - ------------------------- - ---------------------- + I*|---------------------------- - -------------------------| + - - + 3 / --- + --------- + -------------------------
3 _________________ 2 | 2 _________________| 3 _________________ 2 | 2 _________________| 3 \/ 216 1944 _________________
/ _______ | / _______ | / _______ | / _______ | / _______
/ 23 \/ 18849 | / 23 \/ 18849 | / 23 \/ 18849 | / 23 \/ 18849 | / 23 \/ 18849
54*3 / --- + --------- | 54*3 / --- + --------- | 54*3 / --- + --------- | 54*3 / --- + --------- | 27*3 / --- + ---------
\/ 216 1944 \ \/ 216 1944 / \/ 216 1944 \ \/ 216 1944 / \/ 216 1944
$$\left(- \frac{1}{3} + \frac{5}{27 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}\right) + \left(\left(- \frac{1}{3} - \frac{\sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} - \frac{5}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} + \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}\right)\right) + \left(- \frac{1}{3} - \frac{\sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} - \frac{5}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + i \left(- \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2}\right)\right)\right)$$
=
/ _________________ \ / _________________ \
| / _______ | | / _______ |
| ___ / 23 \/ 18849 | | ___ / 23 \/ 18849 |
|\/ 3 *3 / --- + --------- ___ | | \/ 3 *3 / --- + --------- ___ |
| \/ 216 1944 5*\/ 3 | | \/ 216 1944 5*\/ 3 |
-1 + I*|---------------------------- - -------------------------| + I*|- ---------------------------- + -------------------------|
| 2 _________________| | 2 _________________|
| / _______ | | / _______ |
| / 23 \/ 18849 | | / 23 \/ 18849 |
| 54*3 / --- + --------- | | 54*3 / --- + --------- |
\ \/ 216 1944 / \ \/ 216 1944 /
$$-1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} + \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}\right) + i \left(- \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2}\right)$$
producto
/ _________________ / _________________ \\ / _________________ / _________________ \\ | / _______ | / _______ || | / _______ | / _______ || | / 23 \/ 18849 | ___ / 23 \/ 18849 || | / 23 \/ 18849 | ___ / 23 \/ 18849 || / _________________ \ | 3 / --- + --------- | \/ 3 *3 / --- + --------- ___ || | 3 / --- + --------- |\/ 3 *3 / --- + --------- ___ || | / _______ | | 1 5 \/ 216 1944 | \/ 216 1944 5*\/ 3 || | 1 5 \/ 216 1944 | \/ 216 1944 5*\/ 3 || | 1 / 23 \/ 18849 5 | |- - - ------------------------- - ---------------------- + I*|- ---------------------------- + -------------------------||*|- - - ------------------------- - ---------------------- + I*|---------------------------- - -------------------------||*|- - + 3 / --- + --------- + -------------------------| | 3 _________________ 2 | 2 _________________|| | 3 _________________ 2 | 2 _________________|| | 3 \/ 216 1944 _________________| | / _______ | / _______ || | / _______ | / _______ || | / _______ | | / 23 \/ 18849 | / 23 \/ 18849 || | / 23 \/ 18849 | / 23 \/ 18849 || | / 23 \/ 18849 | | 54*3 / --- + --------- | 54*3 / --- + --------- || | 54*3 / --- + --------- | 54*3 / --- + --------- || | 27*3 / --- + --------- | \ \/ 216 1944 \ \/ 216 1944 // \ \/ 216 1944 \ \/ 216 1944 // \ \/ 216 1944 /
$$\left(- \frac{1}{3} - \frac{\sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} - \frac{5}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + i \left(- \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2}\right)\right) \left(- \frac{1}{3} - \frac{\sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} - \frac{5}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}{2} + \frac{5 \sqrt{3}}{54 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}}\right)\right) \left(- \frac{1}{3} + \frac{5}{27 \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}} + \sqrt[3]{\frac{\sqrt{18849}}{1944} + \frac{23}{216}}\right)$$
=
13 -- 36
$$\frac{13}{36}$$
13/36
Respuesta numérica
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x1 = -0.779001641003747 - 0.200761465002314*i
x2 = 0.558003282007493
x3 = -0.779001641003747 + 0.200761465002314*i
x3 = -0.779001641003747 + 0.200761465002314*i