Mister Exam
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How to use it?
- Equation:
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Equation x^2+7x=0
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Equation sin(x)=pi/2
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Equation 5-2/5x=11
- Equation x+y-4=0
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Identical expressions
- x^ three + three *x+ two = zero
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- x3+3*x+2=0
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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 / ___ ___ |
3 \/ 27 + 27*\/ 2 |\/ 3 *\/ 27 + 27*\/ 2 3*\/ 3 |
x1 = - -------------------- + ------------------ + I*|------------------------ + --------------------|
_______________ 6 | 6 _______________|
3 / ___ | 3 / ___ |
2*\/ 27 + 27*\/ 2 \ 2*\/ 27 + 27*\/ 2 /
$$x_{1} = - \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)$$
_______________ / _______________\
3 / ___ | ___ ___ 3 / ___ |
3 \/ 27 + 27*\/ 2 | 3*\/ 3 \/ 3 *\/ 27 + 27*\/ 2 |
x2 = - -------------------- + ------------------ + I*|- -------------------- - ------------------------|
_______________ 6 | _______________ 6 |
3 / ___ | 3 / ___ |
2*\/ 27 + 27*\/ 2 \ 2*\/ 27 + 27*\/ 2 /
$$x_{2} = - \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right)$$
_______________
3 / ___
3 \/ 27 + 27*\/ 2
x3 = ------------------ - ------------------
_______________ 3
3 / ___
\/ 27 + 27*\/ 2
$$x_{3} = - \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} + \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}}$$
x3 = -(27 + 27*sqrt(2))^(1/3)/3 + 3/(27 + 27*sqrt(2))^(1/3)
Sum and product of roots
[src]
sum
_______________ / _______________ \ _______________ / _______________\ _______________
3 / ___ | ___ 3 / ___ ___ | 3 / ___ | ___ ___ 3 / ___ | 3 / ___
3 \/ 27 + 27*\/ 2 |\/ 3 *\/ 27 + 27*\/ 2 3*\/ 3 | 3 \/ 27 + 27*\/ 2 | 3*\/ 3 \/ 3 *\/ 27 + 27*\/ 2 | 3 \/ 27 + 27*\/ 2
- -------------------- + ------------------ + I*|------------------------ + --------------------| + - -------------------- + ------------------ + I*|- -------------------- - ------------------------| + ------------------ - ------------------
_______________ 6 | 6 _______________| _______________ 6 | _______________ 6 | _______________ 3
3 / ___ | 3 / ___ | 3 / ___ | 3 / ___ | 3 / ___
2*\/ 27 + 27*\/ 2 \ 2*\/ 27 + 27*\/ 2 / 2*\/ 27 + 27*\/ 2 \ 2*\/ 27 + 27*\/ 2 / \/ 27 + 27*\/ 2
$$\left(- \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} + \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}}\right) + \left(\left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right)\right) + \left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________ \ | ___ ___ 3 / ___ | | ___ 3 / ___ ___ | | 3*\/ 3 \/ 3 *\/ 27 + 27*\/ 2 | |\/ 3 *\/ 27 + 27*\/ 2 3*\/ 3 | I*|- -------------------- - ------------------------| + I*|------------------------ + --------------------| | _______________ 6 | | 6 _______________| | 3 / ___ | | 3 / ___ | \ 2*\/ 27 + 27*\/ 2 / \ 2*\/ 27 + 27*\/ 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right) + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)$$
product
/ _______________ / _______________ \\ / _______________ / _______________\\ / _______________\ | 3 / ___ | ___ 3 / ___ ___ || | 3 / ___ | ___ ___ 3 / ___ || | 3 / ___ | | 3 \/ 27 + 27*\/ 2 |\/ 3 *\/ 27 + 27*\/ 2 3*\/ 3 || | 3 \/ 27 + 27*\/ 2 | 3*\/ 3 \/ 3 *\/ 27 + 27*\/ 2 || | 3 \/ 27 + 27*\/ 2 | |- -------------------- + ------------------ + I*|------------------------ + --------------------||*|- -------------------- + ------------------ + I*|- -------------------- - ------------------------||*|------------------ - ------------------| | _______________ 6 | 6 _______________|| | _______________ 6 | _______________ 6 || | _______________ 3 | | 3 / ___ | 3 / ___ || | 3 / ___ | 3 / ___ || |3 / ___ | \ 2*\/ 27 + 27*\/ 2 \ 2*\/ 27 + 27*\/ 2 // \ 2*\/ 27 + 27*\/ 2 \ 2*\/ 27 + 27*\/ 2 // \\/ 27 + 27*\/ 2 /
$$\left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)\right) \left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right)\right) \left(- \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} + \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}}\right)$$
=
-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