Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 9x^2+6x+1=0
-
Equation x^4-6x^2+5=0
-
Equation x^3-7x+6=0
- Equation x^4+5*x^2-36=0
- Express {x} in terms of y where:
- 20*x+15*y=1
- 5*x-3*y=8
- 7*x-17*y=14
- -5*x-16*y=9
- Derivative of:
- x^3-3*x+3
- Graphing y =:
- x^3-3*x+3
- Factor polynomial:
- x^3-3*x+3
-
Identical expressions
- x^ three - three *x+ three = zero
- x cubed minus 3 multiply by x plus 3 equally 0
- x to the power of three minus three multiply by x plus three equally zero
- x3-3*x+3=0
- x³-3*x+3=0
- x to the power of 3-3*x+3=0
- x^3-3x+3=0
- x3-3x+3=0
- x^3-3*x+3=O
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Similar expressions
- x^3-3*x-3=0
- x^3+3*x+3=0
x^3-3*x+3=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = 3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = 3$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = 3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = 3$$
Rapid solution
[src]
_______________ / _______________\
/ ___ | / ___ |
/ 81 27*\/ 5 | ___ / 81 27*\/ 5 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
\/ 2 2 3 | 3*\/ 3 \/ 2 2 |
x1 = -------------------- + ---------------------- + I*|- ---------------------- + --------------------------|
6 _______________ | _______________ 6 |
/ ___ | / ___ |
/ 81 27*\/ 5 | / 81 27*\/ 5 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = \frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)$$
_______________ / _______________ \
/ ___ | / ___ |
/ 81 27*\/ 5 | ___ / 81 27*\/ 5 |
3 / -- + -------- | \/ 3 *3 / -- + -------- ___ |
\/ 2 2 3 | \/ 2 2 3*\/ 3 |
x2 = -------------------- + ---------------------- + I*|- -------------------------- + ----------------------|
6 _______________ | 6 _______________|
/ ___ | / ___ |
/ 81 27*\/ 5 | / 81 27*\/ 5 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = \frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)$$
_______________
/ ___
/ 81 27*\/ 5
3 / -- + --------
3 \/ 2 2
x3 = - -------------------- - --------------------
_______________ 3
/ ___
/ 81 27*\/ 5
3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}$$
x3 = -(27*sqrt(5)/2 + 81/2)^(1/3)/3 - 3/(27*sqrt(5)/2 + 81/2)^(1/3)
Sum and product of roots
[src]
sum
_______________ / _______________\ _______________ / _______________ \ _______________
/ ___ | / ___ | / ___ | / ___ | / ___
/ 81 27*\/ 5 | ___ / 81 27*\/ 5 | / 81 27*\/ 5 | ___ / 81 27*\/ 5 | / 81 27*\/ 5
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + -------- | \/ 3 *3 / -- + -------- ___ | 3 / -- + --------
\/ 2 2 3 | 3*\/ 3 \/ 2 2 | \/ 2 2 3 | \/ 2 2 3*\/ 3 | 3 \/ 2 2
-------------------- + ---------------------- + I*|- ---------------------- + --------------------------| + -------------------- + ---------------------- + I*|- -------------------------- + ----------------------| + - -------------------- - --------------------
6 _______________ | _______________ 6 | 6 _______________ | 6 _______________| _______________ 3
/ ___ | / ___ | / ___ | / ___ | / ___
/ 81 27*\/ 5 | / 81 27*\/ 5 | / 81 27*\/ 5 | / 81 27*\/ 5 | / 81 27*\/ 5
2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right) + \left(\left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)\right) + \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________ \ | / ___ | | / ___ | | ___ / 81 27*\/ 5 | | ___ / 81 27*\/ 5 | | ___ \/ 3 *3 / -- + -------- | | \/ 3 *3 / -- + -------- ___ | | 3*\/ 3 \/ 2 2 | | \/ 2 2 3*\/ 3 | I*|- ---------------------- + --------------------------| + I*|- -------------------------- + ----------------------| | _______________ 6 | | 6 _______________| | / ___ | | / ___ | | / 81 27*\/ 5 | | / 81 27*\/ 5 | | 2*3 / -- + -------- | | 2*3 / -- + -------- | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right) + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)$$
product
/ _______________ / _______________\\ / _______________ / _______________ \\ / _______________\ | / ___ | / ___ || | / ___ | / ___ || | / ___ | | / 81 27*\/ 5 | ___ / 81 27*\/ 5 || | / 81 27*\/ 5 | ___ / 81 27*\/ 5 || | / 81 27*\/ 5 | |3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || |3 / -- + -------- | \/ 3 *3 / -- + -------- ___ || | 3 / -- + -------- | |\/ 2 2 3 | 3*\/ 3 \/ 2 2 || |\/ 2 2 3 | \/ 2 2 3*\/ 3 || | 3 \/ 2 2 | |-------------------- + ---------------------- + I*|- ---------------------- + --------------------------||*|-------------------- + ---------------------- + I*|- -------------------------- + ----------------------||*|- -------------------- - --------------------| | 6 _______________ | _______________ 6 || | 6 _______________ | 6 _______________|| | _______________ 3 | | / ___ | / ___ || | / ___ | / ___ || | / ___ | | / 81 27*\/ 5 | / 81 27*\/ 5 || | / 81 27*\/ 5 | / 81 27*\/ 5 || | / 81 27*\/ 5 | | 2*3 / -- + -------- | 2*3 / -- + -------- || | 2*3 / -- + -------- | 2*3 / -- + -------- || | 3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)\right) \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)$$
=
-3
$$-3$$
-3
Numerical answer
[src]
x1 = 1.05190170136777 + 0.565235851677171*i
x2 = 1.05190170136777 - 0.565235851677171*i
x3 = -2.10380340273554
x3 = -2.10380340273554
The graph