Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^3=2
-
Equation 3x^2-2x=0
- Equation x^2+y^2=1
-
Equation x^3=3
- Express {x} in terms of y where:
- -4*x+7*y=-16
- 12*x-20*y=9
- -1*x-20*y=1
- -5*x-12*y=13
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Identical expressions
- x^ three +a*x+b= zero
- x cubed plus a multiply by x plus b equally 0
- x to the power of three plus a multiply by x plus b equally zero
- x3+a*x+b=0
- x³+a*x+b=0
- x to the power of 3+a*x+b=0
- x^3+ax+b=0
- x3+ax+b=0
- x^3+a*x+b=O
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Similar expressions
- x^3-a*x+b=0
- x^3+a*x-b=0
x^3+a*x+b=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = a$$
$$v = \frac{d}{a}$$
$$v = b$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = a$$
$$x_{1} x_{2} x_{3} = b$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = a$$
$$v = \frac{d}{a}$$
$$v = b$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = a$$
$$x_{1} x_{2} x_{3} = b$$
The graph
Rapid solution
[src]
_____________________________
/ _________________
/ / 3 2
/ \/ 108*a + 729*b 27*b
3 / -------------------- + ----
\/ 2 2 a
x1 = - ----------------------------------- + -----------------------------------
3 _____________________________
/ _________________
/ / 3 2
/ \/ 108*a + 729*b 27*b
3 / -------------------- + ----
\/ 2 2
$$x_{1} = \frac{a}{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}$$
_____________________________
/ _________________
/ / 3 2 / ___\
/ \/ 108*a + 729*b 27*b | 1 I*\/ 3 |
3 / -------------------- + ---- *|- - - -------|
\/ 2 2 \ 2 2 / a
x2 = - --------------------------------------------------- + ---------------------------------------------------
3 _____________________________
/ _________________
/ ___\ / / 3 2
| 1 I*\/ 3 | / \/ 108*a + 729*b 27*b
|- - - -------|*3 / -------------------- + ----
\ 2 2 / \/ 2 2
$$x_{2} = \frac{a}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}$$
_____________________________
/ _________________
/ / 3 2 / ___\
/ \/ 108*a + 729*b 27*b | 1 I*\/ 3 |
3 / -------------------- + ---- *|- - + -------|
\/ 2 2 \ 2 2 / a
x3 = - --------------------------------------------------- + ---------------------------------------------------
3 _____________________________
/ _________________
/ ___\ / / 3 2
| 1 I*\/ 3 | / \/ 108*a + 729*b 27*b
|- - + -------|*3 / -------------------- + ----
\ 2 2 / \/ 2 2
$$x_{3} = \frac{a}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}$$
Sum and product of roots
[src]
sum
_____________________________ _____________________________ _____________________________
/ _________________ / _________________ / _________________
/ / 3 2 / / 3 2 / ___\ / / 3 2 / ___\
/ \/ 108*a + 729*b 27*b / \/ 108*a + 729*b 27*b | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b | 1 I*\/ 3 |
3 / -------------------- + ---- 3 / -------------------- + ---- *|- - - -------| 3 / -------------------- + ---- *|- - + -------|
\/ 2 2 a \/ 2 2 \ 2 2 / a \/ 2 2 \ 2 2 / a
0 + - ----------------------------------- + ----------------------------------- + - --------------------------------------------------- + --------------------------------------------------- + - --------------------------------------------------- + ---------------------------------------------------
3 _____________________________ 3 _____________________________ 3 _____________________________
/ _________________ / _________________ / _________________
/ / 3 2 / ___\ / / 3 2 / ___\ / / 3 2
/ \/ 108*a + 729*b 27*b | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b
3 / -------------------- + ---- |- - - -------|*3 / -------------------- + ---- |- - + -------|*3 / -------------------- + ----
\/ 2 2 \ 2 2 / \/ 2 2 \ 2 2 / \/ 2 2
$$\left(\frac{a}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}\right) + \left(\left(\frac{a}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}\right) + \left(\left(\frac{a}{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}\right) + 0\right)\right)$$
=
_____________________________ _____________________________ _____________________________
/ _________________ / _________________ / _________________
/ / 3 2 / / 3 2 / ___\ / / 3 2 / ___\
/ \/ 108*a + 729*b 27*b / \/ 108*a + 729*b 27*b | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b | 1 I*\/ 3 |
3 / -------------------- + ---- 3 / -------------------- + ---- *|- - + -------| 3 / -------------------- + ---- *|- - - -------|
\/ 2 2 a \/ 2 2 \ 2 2 / \/ 2 2 \ 2 2 / a a
- ----------------------------------- + ----------------------------------- - --------------------------------------------------- - --------------------------------------------------- + --------------------------------------------------- + ---------------------------------------------------
3 _____________________________ 3 3 _____________________________ _____________________________
/ _________________ / _________________ / _________________
/ / 3 2 / ___\ / / 3 2 / ___\ / / 3 2
/ \/ 108*a + 729*b 27*b | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b
3 / -------------------- + ---- |- - + -------|*3 / -------------------- + ---- |- - - -------|*3 / -------------------- + ----
\/ 2 2 \ 2 2 / \/ 2 2 \ 2 2 / \/ 2 2
$$\frac{a}{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} + \frac{a}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} + \frac{a}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}$$
product
/ _____________________________ \ / _____________________________ \ / _____________________________ \ | / _________________ | | / _________________ | | / _________________ | | / / 3 2 | | / / 3 2 / ___\ | | / / 3 2 / ___\ | | / \/ 108*a + 729*b 27*b | | / \/ 108*a + 729*b 27*b | 1 I*\/ 3 | | | / \/ 108*a + 729*b 27*b | 1 I*\/ 3 | | | 3 / -------------------- + ---- | | 3 / -------------------- + ---- *|- - - -------| | | 3 / -------------------- + ---- *|- - + -------| | | \/ 2 2 a | | \/ 2 2 \ 2 2 / a | | \/ 2 2 \ 2 2 / a | 1*|- ----------------------------------- + -----------------------------------|*|- --------------------------------------------------- + ---------------------------------------------------|*|- --------------------------------------------------- + ---------------------------------------------------| | 3 _____________________________| | 3 _____________________________| | 3 _____________________________| | / _________________ | | / _________________ | | / _________________ | | / / 3 2 | | / ___\ / / 3 2 | | / ___\ / / 3 2 | | / \/ 108*a + 729*b 27*b | | | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b | | | 1 I*\/ 3 | / \/ 108*a + 729*b 27*b | | 3 / -------------------- + ---- | | |- - - -------|*3 / -------------------- + ---- | | |- - + -------|*3 / -------------------- + ---- | \ \/ 2 2 / \ \ 2 2 / \/ 2 2 / \ \ 2 2 / \/ 2 2 /
$$1 \left(\frac{a}{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}\right) \left(\frac{a}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}\right) \left(\frac{a}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 b}{2} + \frac{\sqrt{108 a^{3} + 729 b^{2}}}{2}}}{3}\right)$$
=
/ ____________________________________ __________________________________\
| _______________ / ______________ / ______________ |
| / 3 2 3 / ___ / 3 2 2/3 3 / ___ / 3 2 |
b*\-72*b - 8*\/ 12*a + 81*b - 3*a*\/ 108*b + 12*\/ 3 *\/ 4*a + 27*b + 3*a*2 *\/ 27*b + 3*\/ 3 *\/ 4*a + 27*b /
---------------------------------------------------------------------------------------------------------------------------------
/ ______________\
| ___ / 3 2 |
8*\9*b + \/ 3 *\/ 4*a + 27*b /
$$\frac{b \left(3 \cdot 2^{\frac{2}{3}} a \sqrt[3]{27 b + 3 \sqrt{3} \sqrt{4 a^{3} + 27 b^{2}}} - 3 a \sqrt[3]{108 b + 12 \sqrt{3} \sqrt{4 a^{3} + 27 b^{2}}} - 72 b - 8 \sqrt{12 a^{3} + 81 b^{2}}\right)}{8 \cdot \left(9 b + \sqrt{3} \sqrt{4 a^{3} + 27 b^{2}}\right)}$$
b*(-72*b - 8*sqrt(12*a^3 + 81*b^2) - 3*a*(108*b + 12*sqrt(3)*sqrt(4*a^3 + 27*b^2))^(1/3) + 3*a*2^(2/3)*(27*b + 3*sqrt(3)*sqrt(4*a^3 + 27*b^2))^(1/3))/(8*(9*b + sqrt(3)*sqrt(4*a^3 + 27*b^2)))