Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x-4)/(x-6)=2
-
Equation e^x=2
-
Equation x^2+4*x=21
-
Equation x^3+x-1=0
- Express {x} in terms of y where:
- 19*x-3*y=-12
- 17*x-7*y=15
- 14*x+3*y=-4
- 14*x-4*y=14
- Factor polynomial:
- x^3+x-1
- Derivative of:
- x^3+x-1
-
Identical expressions
- x^ three +x- one = zero
- x cubed plus x minus 1 equally 0
- x to the power of three plus x minus one equally zero
- x3+x-1=0
- x³+x-1=0
- x to the power of 3+x-1=0
- x^3+x-1=O
-
Similar expressions
- x^3+x+1=0
- x^3-x-1=0
x^3+x-1=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = -1$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = -1$$
Rapid solution
[src]
____________ / ____________ \
/ ____ | / ____ |
/ 1 \/ 93 | ___ / 1 \/ 93 |
3 / - + ------ | \/ 3 *3 / - + ------ ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x1 = - ----------------- + ------------------- + I*|- ----------------------- - -------------------|
2 ____________ | 2 ____________|
/ ____ | / ____ |
/ 1 \/ 93 | / 1 \/ 93 |
6*3 / - + ------ | 6*3 / - + ------ |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right)$$
____________ / ____________ \
/ ____ | / ____ |
/ 1 \/ 93 | ___ / 1 \/ 93 |
3 / - + ------ |\/ 3 *3 / - + ------ ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x2 = - ----------------- + ------------------- + I*|----------------------- + -------------------|
2 ____________ | 2 ____________|
/ ____ | / ____ |
/ 1 \/ 93 | / 1 \/ 93 |
6*3 / - + ------ | 6*3 / - + ------ |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)$$
____________
/ ____
/ 1 \/ 93 1
x3 = 3 / - + ------ - -------------------
\/ 2 18 ____________
/ ____
/ 1 \/ 93
3*3 / - + ------
\/ 2 18
$$x_{3} = - \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}$$
x3 = -1/(3*(1/2 + sqrt(93)/18)^(1/3)) + (1/2 + sqrt(93)/18)^(1/3)
Sum and product of roots
[src]
sum
____________ / ____________ \ ____________ / ____________ \
/ ____ | / ____ | / ____ | / ____ |
/ 1 \/ 93 | ___ / 1 \/ 93 | / 1 \/ 93 | ___ / 1 \/ 93 | ____________
3 / - + ------ | \/ 3 *3 / - + ------ ___ | 3 / - + ------ |\/ 3 *3 / - + ------ ___ | / ____
\/ 2 18 1 | \/ 2 18 \/ 3 | \/ 2 18 1 | \/ 2 18 \/ 3 | / 1 \/ 93 1
- ----------------- + ------------------- + I*|- ----------------------- - -------------------| + - ----------------- + ------------------- + I*|----------------------- + -------------------| + 3 / - + ------ - -------------------
2 ____________ | 2 ____________| 2 ____________ | 2 ____________| \/ 2 18 ____________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 1 \/ 93 | / 1 \/ 93 | / 1 \/ 93 | / 1 \/ 93 | / 1 \/ 93
6*3 / - + ------ | 6*3 / - + ------ | 6*3 / - + ------ | 6*3 / - + ------ | 3*3 / - + ------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(- \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)\right)\right)$$
=
/ ____________ \ / ____________ \ | / ____ | | / ____ | | ___ / 1 \/ 93 | | ___ / 1 \/ 93 | |\/ 3 *3 / - + ------ ___ | | \/ 3 *3 / - + ------ ___ | | \/ 2 18 \/ 3 | | \/ 2 18 \/ 3 | I*|----------------------- + -------------------| + I*|- ----------------------- - -------------------| | 2 ____________| | 2 ____________| | / ____ | | / ____ | | / 1 \/ 93 | | / 1 \/ 93 | | 6*3 / - + ------ | | 6*3 / - + ------ | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right) + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)$$
product
/ ____________ / ____________ \\ / ____________ / ____________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 1 \/ 93 | ___ / 1 \/ 93 || | / 1 \/ 93 | ___ / 1 \/ 93 || / ____________ \ | 3 / - + ------ | \/ 3 *3 / - + ------ ___ || | 3 / - + ------ |\/ 3 *3 / - + ------ ___ || | / ____ | | \/ 2 18 1 | \/ 2 18 \/ 3 || | \/ 2 18 1 | \/ 2 18 \/ 3 || | / 1 \/ 93 1 | |- ----------------- + ------------------- + I*|- ----------------------- - -------------------||*|- ----------------- + ------------------- + I*|----------------------- + -------------------||*|3 / - + ------ - -------------------| | 2 ____________ | 2 ____________|| | 2 ____________ | 2 ____________|| |\/ 2 18 ____________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 1 \/ 93 | / 1 \/ 93 || | / 1 \/ 93 | / 1 \/ 93 || | / 1 \/ 93 | | 6*3 / - + ------ | 6*3 / - + ------ || | 6*3 / - + ------ | 6*3 / - + ------ || | 3*3 / - + ------ | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right)\right) \left(- \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}\right)$$
=
1
$$1$$
1
Numerical answer
[src]
x1 = -0.34116390191401 + 1.16154139999725*i
x2 = 0.682327803828019
x3 = -0.34116390191401 - 1.16154139999725*i
x3 = -0.34116390191401 - 1.16154139999725*i
The graph