Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation sin(x)=4
-
Equation 5^x=125
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Equation (|x+4|)=5
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Equation 2*sin(x)=1
- Express {x} in terms of y where:
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
- -5*x-2*y=9
- Graphing y =:
- x^3+2x+2
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Identical expressions
- x^ three + two x+2= zero
- x cubed plus 2x plus 2 equally 0
- x to the power of three plus two x plus 2 equally zero
- x3+2x+2=0
- x³+2x+2=0
- x to the power of 3+2x+2=0
- x^3+2x+2=O
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Similar expressions
- x^3-2x+2=0
- x^3+2x-2=0
x^3+2x+2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = 2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = 2$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = 2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = 2$$
Sum and product of roots
[src]
sum
________________ / ________________\ ________________ / ________________\ ________________
3 / _____ | ___ ___ 3 / _____ | 3 / _____ | ___ ___ 3 / _____ | 3 / _____
1 \/ 27 + 3*\/ 105 | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 | 1 \/ 27 + 3*\/ 105 | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 | 2 \/ 27 + 3*\/ 105
0 + - ------------------- + ------------------- + I*|------------------- + -------------------------| + - ------------------- + ------------------- + I*|- ------------------- - -------------------------| + ------------------- - -------------------
________________ 6 | ________________ 6 | ________________ 6 | ________________ 6 | ________________ 3
3 / _____ |3 / _____ | 3 / _____ | 3 / _____ | 3 / _____
\/ 27 + 3*\/ 105 \\/ 27 + 3*\/ 105 / \/ 27 + 3*\/ 105 \ \/ 27 + 3*\/ 105 / \/ 27 + 3*\/ 105
$$\left(- \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{3} + \frac{2}{\sqrt[3]{27 + 3 \sqrt{105}}}\right) + \left(\left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)\right) + \left(0 + \left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)\right)\right)\right)$$
=
/ ________________\ / ________________\ | ___ ___ 3 / _____ | | ___ ___ 3 / _____ | | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 | | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 | I*|------------------- + -------------------------| + I*|- ------------------- - -------------------------| | ________________ 6 | | ________________ 6 | |3 / _____ | | 3 / _____ | \\/ 27 + 3*\/ 105 / \ \/ 27 + 3*\/ 105 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right) + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)$$
product
/ ________________ / ________________\\ / ________________ / ________________\\ / ________________\ | 3 / _____ | ___ ___ 3 / _____ || | 3 / _____ | ___ ___ 3 / _____ || | 3 / _____ | | 1 \/ 27 + 3*\/ 105 | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 || | 1 \/ 27 + 3*\/ 105 | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 || | 2 \/ 27 + 3*\/ 105 | 1*|- ------------------- + ------------------- + I*|------------------- + -------------------------||*|- ------------------- + ------------------- + I*|- ------------------- - -------------------------||*|------------------- - -------------------| | ________________ 6 | ________________ 6 || | ________________ 6 | ________________ 6 || | ________________ 3 | | 3 / _____ |3 / _____ || | 3 / _____ | 3 / _____ || |3 / _____ | \ \/ 27 + 3*\/ 105 \\/ 27 + 3*\/ 105 // \ \/ 27 + 3*\/ 105 \ \/ 27 + 3*\/ 105 // \\/ 27 + 3*\/ 105 /
$$1 \left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)\right) \left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)\right) \left(- \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{3} + \frac{2}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)$$
=
-2
$$-2$$
-2
Rapid solution
[src]
________________ / ________________\
3 / _____ | ___ ___ 3 / _____ |
1 \/ 27 + 3*\/ 105 | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 |
x1 = - ------------------- + ------------------- + I*|------------------- + -------------------------|
________________ 6 | ________________ 6 |
3 / _____ |3 / _____ |
\/ 27 + 3*\/ 105 \\/ 27 + 3*\/ 105 /
$$x_{1} = - \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)$$
________________ / ________________\
3 / _____ | ___ ___ 3 / _____ |
1 \/ 27 + 3*\/ 105 | \/ 3 \/ 3 *\/ 27 + 3*\/ 105 |
x2 = - ------------------- + ------------------- + I*|- ------------------- - -------------------------|
________________ 6 | ________________ 6 |
3 / _____ | 3 / _____ |
\/ 27 + 3*\/ 105 \ \/ 27 + 3*\/ 105 /
$$x_{2} = - \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)$$
________________
3 / _____
2 \/ 27 + 3*\/ 105
x3 = ------------------- - -------------------
________________ 3
3 / _____
\/ 27 + 3*\/ 105
$$x_{3} = - \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{3} + \frac{2}{\sqrt[3]{27 + 3 \sqrt{105}}}$$
Numerical answer
[src]
x1 = 0.385458498529624 + 1.56388451052696*i
x2 = 0.385458498529624 - 1.56388451052696*i
x3 = -0.770916997059248
x3 = -0.770916997059248
The graph