Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 9^x-8*3^x-9=0
- Equation x+y=4
-
Equation 5x+8=-12
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Equation x/9+x=-3
- Express {x} in terms of y where:
- 18*x+17*y=-14
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
- Graphing y =:
- x^3-6*x+2
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Identical expressions
- x^ three - six *x+ two = zero
- x cubed minus 6 multiply by x plus 2 equally 0
- x to the power of three minus six multiply by x plus two equally zero
- x3-6*x+2=0
- x³-6*x+2=0
- x to the power of 3-6*x+2=0
- x^3-6x+2=0
- x3-6x+2=0
- x^3-6*x+2=O
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Similar expressions
- x^3-6*x-2=0
- x^3+6*x+2=0
x^3-6*x+2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -6$$
$$v = \frac{d}{a}$$
$$v = 2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -6$$
$$x_{1} x_{2} x_{3} = 2$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -6$$
$$v = \frac{d}{a}$$
$$v = 2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -6$$
$$x_{1} x_{2} x_{3} = 2$$
Rapid solution
[src]
/ / / ___\\ / / ___\\\ / / ___\\ / / ___\\
| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|
| \/ 2 *sin|-----------| \/ 6 *cos|-----------|| \/ 2 *cos|-----------| \/ 6 *sin|-----------|
/ 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 /
x1 = - 6*re|------------------------------------| + I*|- 6*im|------------------------------------| + ---------------------- + ----------------------| + ---------------------- - ----------------------
| _________________ / ___\| | | _________________ / ___\| 2 2 | 2 2
|3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || |
|\/ 27 + 27*I*\/ 7 *|- - - -------|| | |\/ 27 + 27*I*\/ 7 *|- - - -------|| |
\ \ 2 2 // \ \ \ 2 2 // /
$$x_{1} = - \frac{\sqrt{6} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + i \left(- 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2}\right)$$
/ / / ___\\ / / ___\\\ / / ___\\ / / ___\\
| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|
| \/ 2 *sin|-----------| \/ 6 *cos|-----------|| \/ 2 *cos|-----------| \/ 6 *sin|-----------|
/ 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 /
x2 = - 6*re|------------------------------------| + I*|- 6*im|------------------------------------| + ---------------------- - ----------------------| + ---------------------- + ----------------------
| _________________ / ___\| | | _________________ / ___\| 2 2 | 2 2
|3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || |
|\/ 27 + 27*I*\/ 7 *|- - + -------|| | |\/ 27 + 27*I*\/ 7 *|- - + -------|| |
\ \ 2 2 // \ \ \ 2 2 // /
$$x_{2} = \frac{\sqrt{6} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + i \left(- \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)}\right)$$
_________________
3 / ___
6 \/ 27 + 27*I*\/ 7
x3 = - -------------------- - --------------------
_________________ 3
3 / ___
\/ 27 + 27*I*\/ 7
$$x_{3} = - \frac{\sqrt[3]{27 + 27 \sqrt{7} i}}{3} - \frac{6}{\sqrt[3]{27 + 27 \sqrt{7} i}}$$
x3 = -(27 + 27*sqrt(7)*i)^(1/3)/3 - 6/(27 + 27*sqrt(7)*i)^(1/3)
Sum and product of roots
[src]
sum
/ / / ___\\ / / ___\\\ / / ___\\ / / ___\\ / / / ___\\ / / ___\\\ / / ___\\ / / ___\\
| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /| | ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /| _________________
| \/ 2 *sin|-----------| \/ 6 *cos|-----------|| \/ 2 *cos|-----------| \/ 6 *sin|-----------| | \/ 2 *sin|-----------| \/ 6 *cos|-----------|| \/ 2 *cos|-----------| \/ 6 *sin|-----------| 3 / ___
/ 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 / / 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 / 6 \/ 27 + 27*I*\/ 7
- 6*re|------------------------------------| + I*|- 6*im|------------------------------------| + ---------------------- + ----------------------| + ---------------------- - ---------------------- + - 6*re|------------------------------------| + I*|- 6*im|------------------------------------| + ---------------------- - ----------------------| + ---------------------- + ---------------------- + - -------------------- - --------------------
| _________________ / ___\| | | _________________ / ___\| 2 2 | 2 2 | _________________ / ___\| | | _________________ / ___\| 2 2 | 2 2 _________________ 3
|3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | 3 / ___
|\/ 27 + 27*I*\/ 7 *|- - - -------|| | |\/ 27 + 27*I*\/ 7 *|- - - -------|| | |\/ 27 + 27*I*\/ 7 *|- - + -------|| | |\/ 27 + 27*I*\/ 7 *|- - + -------|| | \/ 27 + 27*I*\/ 7
\ \ 2 2 // \ \ \ 2 2 // / \ \ 2 2 // \ \ \ 2 2 // /
$$\left(\left(\frac{\sqrt{6} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + i \left(- \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)}\right)\right) + \left(- \frac{\sqrt{6} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + i \left(- 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2}\right)\right)\right) + \left(- \frac{\sqrt[3]{27 + 27 \sqrt{7} i}}{3} - \frac{6}{\sqrt[3]{27 + 27 \sqrt{7} i}}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
_________________ | ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| | ___ |atan\\/ 7 /| ___ |atan\\/ 7 /||
3 / ___ | \/ 2 *sin|-----------| \/ 6 *cos|-----------|| | \/ 2 *sin|-----------| \/ 6 *cos|-----------|| / / ___\\
6 / 1 \ / 1 \ \/ 27 + 27*I*\/ 7 | / 1 \ \ 3 / \ 3 /| | / 1 \ \ 3 / \ 3 /| ___ |atan\\/ 7 /|
- -------------------- - 6*re|------------------------------------| - 6*re|------------------------------------| - -------------------- + I*|- 6*im|------------------------------------| + ---------------------- - ----------------------| + I*|- 6*im|------------------------------------| + ---------------------- + ----------------------| + \/ 2 *cos|-----------|
_________________ | _________________ / ___\| | _________________ / ___\| 3 | | _________________ / ___\| 2 2 | | | _________________ / ___\| 2 2 | \ 3 /
3 / ___ |3 / ___ | 1 I*\/ 3 || |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | | |3 / ___ | 1 I*\/ 3 || |
\/ 27 + 27*I*\/ 7 |\/ 27 + 27*I*\/ 7 *|- - + -------|| |\/ 27 + 27*I*\/ 7 *|- - - -------|| | |\/ 27 + 27*I*\/ 7 *|- - + -------|| | | |\/ 27 + 27*I*\/ 7 *|- - - -------|| |
\ \ 2 2 // \ \ 2 2 // \ \ \ 2 2 // / \ \ \ 2 2 // /
$$- 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)} - \frac{\sqrt[3]{27 + 27 \sqrt{7} i}}{3} + i \left(- 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)}\right) - \frac{6}{\sqrt[3]{27 + 27 \sqrt{7} i}}$$
product
/ / / / ___\\ / / ___\\\ / / ___\\ / / ___\\\ / / / / ___\\ / / ___\\\ / / ___\\ / / ___\\\ | | ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| | | ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| ___ |atan\\/ 7 /| ___ |atan\\/ 7 /|| / _________________\ | | \/ 2 *sin|-----------| \/ 6 *cos|-----------|| \/ 2 *cos|-----------| \/ 6 *sin|-----------|| | | \/ 2 *sin|-----------| \/ 6 *cos|-----------|| \/ 2 *cos|-----------| \/ 6 *sin|-----------|| | 3 / ___ | | / 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 /| | / 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 /| | 6 \/ 27 + 27*I*\/ 7 | |- 6*re|------------------------------------| + I*|- 6*im|------------------------------------| + ---------------------- + ----------------------| + ---------------------- - ----------------------|*|- 6*re|------------------------------------| + I*|- 6*im|------------------------------------| + ---------------------- - ----------------------| + ---------------------- + ----------------------|*|- -------------------- - --------------------| | | _________________ / ___\| | | _________________ / ___\| 2 2 | 2 2 | | | _________________ / ___\| | | _________________ / ___\| 2 2 | 2 2 | | _________________ 3 | | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | | | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | | | 3 / ___ | | |\/ 27 + 27*I*\/ 7 *|- - - -------|| | |\/ 27 + 27*I*\/ 7 *|- - - -------|| | | | |\/ 27 + 27*I*\/ 7 *|- - + -------|| | |\/ 27 + 27*I*\/ 7 *|- - + -------|| | | \ \/ 27 + 27*I*\/ 7 / \ \ \ 2 2 // \ \ \ 2 2 // / / \ \ \ 2 2 // \ \ \ 2 2 // / /
$$\left(- \frac{\sqrt{6} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + i \left(- 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2}\right)\right) \left(\frac{\sqrt{6} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)} + i \left(- \frac{\sqrt{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} + \frac{\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}}{2} - 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{27 + 27 \sqrt{7} i}}\right)}\right)\right) \left(- \frac{\sqrt[3]{27 + 27 \sqrt{7} i}}{3} - \frac{6}{\sqrt[3]{27 + 27 \sqrt{7} i}}\right)$$
=
/ / / ___\\\ / 2/3\
| |2*atan\\/ 7 /|| | / ___\ |
2*|1 - 2*cos|-------------||*\2 + \1 + I*\/ 7 / /
\ \ 3 //
---------------------------------------------------
_____________
3 / ___
\/ 1 + I*\/ 7
$$\frac{2 \left(1 - 2 \cos{\left(\frac{2 \operatorname{atan}{\left(\sqrt{7} \right)}}{3} \right)}\right) \left(2 + \left(1 + \sqrt{7} i\right)^{\frac{2}{3}}\right)}{\sqrt[3]{1 + \sqrt{7} i}}$$
2*(1 - 2*cos(2*atan(sqrt(7))/3))*(2 + (1 + i*sqrt(7))^(2/3))/(1 + i*sqrt(7))^(1/3)
Numerical answer
[src]
x1 = -2.60167913188315
x2 = 0.339876886623183
x3 = 2.26180224525997
x3 = 2.26180224525997
The graph