Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2+4*x+3=0
-
Equation -x=14
-
Equation x^4-10x^2+9=0
-
Equation 6x+2=-1
- Express {x} in terms of y where:
- 15*x+1*y=19
- -2*x+2*y=4
- -17*x-8*y=7
- -20*x+14*y=16
- Graphing y =:
- x^3+3*x-2
-
Identical expressions
- x^ three + three *x- two = zero
- x cubed plus 3 multiply by x minus 2 equally 0
- x to the power of three plus three multiply by x minus two equally zero
- x3+3*x-2=0
- x³+3*x-2=0
- x to the power of 3+3*x-2=0
- x^3+3x-2=0
- x3+3x-2=0
- x^3+3*x-2=O
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Similar expressions
- x^3+3*x+2=0
- x^3-3*x-2=0
x^3+3*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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = -2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = -2$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = -2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = -2$$
Rapid solution
[src]
___________ / ___________\
3 / ___ | ___ ___ 3 / ___ |
1 \/ 1 + \/ 2 | \/ 3 \/ 3 *\/ 1 + \/ 2 |
x1 = ---------------- - -------------- + I*|- ---------------- - --------------------|
___________ 2 | ___________ 2 |
3 / ___ | 3 / ___ |
2*\/ 1 + \/ 2 \ 2*\/ 1 + \/ 2 /
$$x_{1} = - \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right)$$
___________ / ___________\
3 / ___ | ___ ___ 3 / ___ |
1 \/ 1 + \/ 2 | \/ 3 \/ 3 *\/ 1 + \/ 2 |
x2 = ---------------- - -------------- + I*|---------------- + --------------------|
___________ 2 | ___________ 2 |
3 / ___ | 3 / ___ |
2*\/ 1 + \/ 2 \2*\/ 1 + \/ 2 /
$$x_{2} = - \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)$$
___________
3 / ___ 1
x3 = \/ 1 + \/ 2 - --------------
___________
3 / ___
\/ 1 + \/ 2
$$x_{3} = - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}$$
x3 = -1/(1 + sqrt(2))^(1/3) + (1 + sqrt(2))^(1/3)
Sum and product of roots
[src]
sum
___________ / ___________\ ___________ / ___________\
3 / ___ | ___ ___ 3 / ___ | 3 / ___ | ___ ___ 3 / ___ | ___________
1 \/ 1 + \/ 2 | \/ 3 \/ 3 *\/ 1 + \/ 2 | 1 \/ 1 + \/ 2 | \/ 3 \/ 3 *\/ 1 + \/ 2 | 3 / ___ 1
---------------- - -------------- + I*|- ---------------- - --------------------| + ---------------- - -------------- + I*|---------------- + --------------------| + \/ 1 + \/ 2 - --------------
___________ 2 | ___________ 2 | ___________ 2 | ___________ 2 | ___________
3 / ___ | 3 / ___ | 3 / ___ | 3 / ___ | 3 / ___
2*\/ 1 + \/ 2 \ 2*\/ 1 + \/ 2 / 2*\/ 1 + \/ 2 \2*\/ 1 + \/ 2 / \/ 1 + \/ 2
$$\left(- \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}\right) + \left(\left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right)\right) + \left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)\right)\right)$$
=
/ ___________\ / ___________\ | ___ ___ 3 / ___ | | ___ ___ 3 / ___ | | \/ 3 \/ 3 *\/ 1 + \/ 2 | | \/ 3 \/ 3 *\/ 1 + \/ 2 | I*|---------------- + --------------------| + I*|- ---------------- - --------------------| | ___________ 2 | | ___________ 2 | | 3 / ___ | | 3 / ___ | \2*\/ 1 + \/ 2 / \ 2*\/ 1 + \/ 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)$$
product
/ ___________ / ___________\\ / ___________ / ___________\\ | 3 / ___ | ___ ___ 3 / ___ || | 3 / ___ | ___ ___ 3 / ___ || / ___________ \ | 1 \/ 1 + \/ 2 | \/ 3 \/ 3 *\/ 1 + \/ 2 || | 1 \/ 1 + \/ 2 | \/ 3 \/ 3 *\/ 1 + \/ 2 || |3 / ___ 1 | |---------------- - -------------- + I*|- ---------------- - --------------------||*|---------------- - -------------- + I*|---------------- + --------------------||*|\/ 1 + \/ 2 - --------------| | ___________ 2 | ___________ 2 || | ___________ 2 | ___________ 2 || | ___________| | 3 / ___ | 3 / ___ || | 3 / ___ | 3 / ___ || | 3 / ___ | \2*\/ 1 + \/ 2 \ 2*\/ 1 + \/ 2 // \2*\/ 1 + \/ 2 \2*\/ 1 + \/ 2 // \ \/ 1 + \/ 2 /
$$\left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right)\right) \left(- \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}\right)$$
=
2
$$2$$
2
Numerical answer
[src]
x1 = -0.298035818991661 - 1.80733949445202*i
x2 = -0.298035818991661 + 1.80733949445202*i
x3 = 0.596071637983322
x3 = 0.596071637983322
The graph