Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 6x+1=-4x
-
Equation 2*sin(x)-1=0
-
Equation -33-y=-19
- Equation x^2+y^2=9
- Express {x} in terms of y where:
- 10*x-18*y=3
- -14*x-12*y=5
- 13*x-12*y=19
- -11*x-3*y=14
- Graphing y =:
- x^3-x^2-x-1
-
Identical expressions
- x^ three -x^ two -x- one = zero
- x cubed minus x squared minus x minus 1 equally 0
- x to the power of three minus x to the power of two minus x minus one equally zero
- x3-x2-x-1=0
- x³-x²-x-1=0
- x to the power of 3-x to the power of 2-x-1=0
- x^3-x^2-x-1=O
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Similar expressions
- x^3-x^2-x+1=0
- x^3+x^2-x-1=0
- x^3-x^2+x-1=0
x^3-x^2-x-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
3 2 x - x - x - 1 = 0
$$\left(- x + \left(x^{3} - x^{2}\right)\right) - 1 = 0$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -1$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -1$$
Sum and product of roots
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sum
_____________ / _____________ \ _____________ / _____________ \
/ ____ | / ____ | / ____ | / ____ |
/ 19 \/ 33 | ___ / 19 \/ 33 | / 19 \/ 33 | ___ / 19 \/ 33 | _____________
3 / -- + ------ | \/ 3 *3 / -- + ------ ___ | 3 / -- + ------ |\/ 3 *3 / -- + ------ ___ | / ____
1 2 \/ 27 9 | \/ 27 9 2*\/ 3 | 1 2 \/ 27 9 | \/ 27 9 2*\/ 3 | 1 / 19 \/ 33 4
- - -------------------- - ------------------ + I*|- ------------------------ + --------------------| + - - -------------------- - ------------------ + I*|------------------------ - --------------------| + - + 3 / -- + ------ + --------------------
3 _____________ 2 | 2 _____________| 3 _____________ 2 | 2 _____________| 3 \/ 27 9 _____________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 19 \/ 33 | / 19 \/ 33 | / 19 \/ 33 | / 19 \/ 33 | / 19 \/ 33
9*3 / -- + ------ | 9*3 / -- + ------ | 9*3 / -- + ------ | 9*3 / -- + ------ | 9*3 / -- + ------
\/ 27 9 \ \/ 27 9 / \/ 27 9 \ \/ 27 9 / \/ 27 9
$$\left(\frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{1}{3} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2}\right)\right)\right)$$
=
/ _____________ \ / _____________ \
| / ____ | | / ____ |
| ___ / 19 \/ 33 | | ___ / 19 \/ 33 |
|\/ 3 *3 / -- + ------ ___ | | \/ 3 *3 / -- + ------ ___ |
| \/ 27 9 2*\/ 3 | | \/ 27 9 2*\/ 3 |
1 + I*|------------------------ - --------------------| + I*|- ------------------------ + --------------------|
| 2 _____________| | 2 _____________|
| / ____ | | / ____ |
| / 19 \/ 33 | | / 19 \/ 33 |
| 9*3 / -- + ------ | | 9*3 / -- + ------ |
\ \/ 27 9 / \ \/ 27 9 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}\right) + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2}\right)$$
product
/ _____________ / _____________ \\ / _____________ / _____________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 19 \/ 33 | ___ / 19 \/ 33 || | / 19 \/ 33 | ___ / 19 \/ 33 || / _____________ \ | 3 / -- + ------ | \/ 3 *3 / -- + ------ ___ || | 3 / -- + ------ |\/ 3 *3 / -- + ------ ___ || | / ____ | |1 2 \/ 27 9 | \/ 27 9 2*\/ 3 || |1 2 \/ 27 9 | \/ 27 9 2*\/ 3 || |1 / 19 \/ 33 4 | |- - -------------------- - ------------------ + I*|- ------------------------ + --------------------||*|- - -------------------- - ------------------ + I*|------------------------ - --------------------||*|- + 3 / -- + ------ + --------------------| |3 _____________ 2 | 2 _____________|| |3 _____________ 2 | 2 _____________|| |3 \/ 27 9 _____________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 19 \/ 33 | / 19 \/ 33 || | / 19 \/ 33 | / 19 \/ 33 || | / 19 \/ 33 | | 9*3 / -- + ------ | 9*3 / -- + ------ || | 9*3 / -- + ------ | 9*3 / -- + ------ || | 9*3 / -- + ------ | \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{1}{3} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}\right)\right) \left(\frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}\right)$$
=
1
$$1$$
1
Rapid solution
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_____________ / _____________ \
/ ____ | / ____ |
/ 19 \/ 33 | ___ / 19 \/ 33 |
3 / -- + ------ | \/ 3 *3 / -- + ------ ___ |
1 2 \/ 27 9 | \/ 27 9 2*\/ 3 |
x1 = - - -------------------- - ------------------ + I*|- ------------------------ + --------------------|
3 _____________ 2 | 2 _____________|
/ ____ | / ____ |
/ 19 \/ 33 | / 19 \/ 33 |
9*3 / -- + ------ | 9*3 / -- + ------ |
\/ 27 9 \ \/ 27 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}\right)$$
_____________ / _____________ \
/ ____ | / ____ |
/ 19 \/ 33 | ___ / 19 \/ 33 |
3 / -- + ------ |\/ 3 *3 / -- + ------ ___ |
1 2 \/ 27 9 | \/ 27 9 2*\/ 3 |
x2 = - - -------------------- - ------------------ + I*|------------------------ - --------------------|
3 _____________ 2 | 2 _____________|
/ ____ | / ____ |
/ 19 \/ 33 | / 19 \/ 33 |
9*3 / -- + ------ | 9*3 / -- + ------ |
\/ 27 9 \ \/ 27 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{1}{3} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}}{2}\right)$$
_____________
/ ____
1 / 19 \/ 33 4
x3 = - + 3 / -- + ------ + --------------------
3 \/ 27 9 _____________
/ ____
/ 19 \/ 33
9*3 / -- + ------
\/ 27 9
$$x_{3} = \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}} + \sqrt[3]{\frac{\sqrt{33}}{9} + \frac{19}{27}}$$
x3 = 1/3 + 4/(9*(sqrt(33)/9 + 19/27)^(1/3)) + (sqrt(33)/9 + 19/27)^(1/3)
Numerical answer
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x1 = -0.419643377607081 - 0.606290729207199*i
x2 = -0.419643377607081 + 0.606290729207199*i
x3 = 1.83928675521416
x3 = 1.83928675521416
The graph