Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation (x+2)^4-4*(x+2)^2-5=0
-
Equation 12/(x+3)=-6/7
- Equation (x+20)*(-x+10)=0
-
Equation x+970=69*32
- Express {x} in terms of y where:
- -12*x-8*y=13
- -4*x+19*y=-13
- -7*x+15*y=18
- -11*x+10*y=-3
- Factor polynomial:
- x^3+x+1
-
Identical expressions
- x^ three +x+ one = zero
- x cubed plus x plus 1 equally 0
- x to the power of three plus x plus one equally zero
- x3+x+1=0
- x³+x+1=0
- x to the power of 3+x+1=0
- x^3+x+1=O
-
Similar expressions
- x^3-x+1=0
- x^3+x-1=0
x^3+x+1=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = 1$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = 1$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = 1$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = 1$$
Sum and product of roots
[src]
sum
_______________ / _______________\ _______________ / _______________\ _______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 27 3*\/ 93 | ___ / 27 3*\/ 93 | / 27 3*\/ 93 | ___ / 27 3*\/ 93 | / 27 3*\/ 93
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + --------
1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2
- ---------------------- + -------------------- + I*|---------------------- + --------------------------| + - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------| + -------------------- - --------------------
_______________ 6 | _______________ 6 | _______________ 6 | _______________ 6 | _______________ 3
/ ____ | / ____ | / ____ | / ____ | / ____
/ 27 3*\/ 93 | / 27 3*\/ 93 | / 27 3*\/ 93 | / 27 3*\/ 93 | / 27 3*\/ 93
2*3 / -- + -------- |2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right) + \left(\left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)\right) + \left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________\ | / ____ | | / ____ | | ___ / 27 3*\/ 93 | | ___ / 27 3*\/ 93 | | ___ \/ 3 *3 / -- + -------- | | ___ \/ 3 *3 / -- + -------- | | \/ 3 \/ 2 2 | | \/ 3 \/ 2 2 | I*|---------------------- + --------------------------| + I*|- ---------------------- - --------------------------| | _______________ 6 | | _______________ 6 | | / ____ | | / ____ | | / 27 3*\/ 93 | | / 27 3*\/ 93 | |2*3 / -- + -------- | | 2*3 / -- + -------- | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)$$
product
/ _______________ / _______________\\ / _______________ / _______________\\ / _______________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 27 3*\/ 93 | ___ / 27 3*\/ 93 || | / 27 3*\/ 93 | ___ / 27 3*\/ 93 || | / 27 3*\/ 93 | | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- | | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | |- ---------------------- + -------------------- + I*|---------------------- + --------------------------||*|- ---------------------- + -------------------- + I*|- ---------------------- - --------------------------||*|-------------------- - --------------------| | _______________ 6 | _______________ 6 || | _______________ 6 | _______________ 6 || | _______________ 3 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 27 3*\/ 93 | / 27 3*\/ 93 || | / 27 3*\/ 93 | / 27 3*\/ 93 || | / 27 3*\/ 93 | | 2*3 / -- + -------- |2*3 / -- + -------- || | 2*3 / -- + -------- | 2*3 / -- + -------- || |3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \\/ 2 2 /
$$\left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)\right) \left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)$$
=
_________________ _______________ 2/3 _________________ 2/3 _______________ 2/3 2/3
3 / ____ 2/3 3 / ____ 3 ___ / ____\ ____ 3 / ____ 3 ____ / ____\ 2/3 ____ 3 / ____ 3 ___ ____ / ____\ 3 ___ 6 ___ ____ / ____\
9*\/ 108 + 12*\/ 93 9*2 *\/ 27 + 3*\/ 93 \/ 2 *\27 + 3*\/ 93 / \/ 93 *\/ 108 + 12*\/ 93 \/ 18 *\9 + \/ 93 / 2 *\/ 93 *\/ 27 + 3*\/ 93 \/ 2 *\/ 93 *\27 + 3*\/ 93 / \/ 2 *\/ 3 *\/ 31 *\9 + \/ 93 /
-1 + ---------------------- - ------------------------- - ------------------------ - --------------------------- + ---------------------- + ------------------------------ + ------------------------------- - ----------------------------------
32 32 8 32 8 32 72 24
$$- \frac{\sqrt[3]{2} \sqrt[6]{3} \sqrt{31} \left(9 + \sqrt{93}\right)^{\frac{2}{3}}}{24} - \frac{\sqrt[3]{2} \left(27 + 3 \sqrt{93}\right)^{\frac{2}{3}}}{8} - \frac{\sqrt{93} \sqrt[3]{108 + 12 \sqrt{93}}}{32} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt[3]{27 + 3 \sqrt{93}}}{32} - 1 + \frac{9 \sqrt[3]{108 + 12 \sqrt{93}}}{32} + \frac{2^{\frac{2}{3}} \sqrt{93} \sqrt[3]{27 + 3 \sqrt{93}}}{32} + \frac{\sqrt[3]{18} \left(9 + \sqrt{93}\right)^{\frac{2}{3}}}{8} + \frac{\sqrt[3]{2} \sqrt{93} \left(27 + 3 \sqrt{93}\right)^{\frac{2}{3}}}{72}$$
-1 + 9*(108 + 12*sqrt(93))^(1/3)/32 - 9*2^(2/3)*(27 + 3*sqrt(93))^(1/3)/32 - 2^(1/3)*(27 + 3*sqrt(93))^(2/3)/8 - sqrt(93)*(108 + 12*sqrt(93))^(1/3)/32 + 18^(1/3)*(9 + sqrt(93))^(2/3)/8 + 2^(2/3)*sqrt(93)*(27 + 3*sqrt(93))^(1/3)/32 + 2^(1/3)*sqrt(93)*(27 + 3*sqrt(93))^(2/3)/72 - 2^(1/3)*3^(1/6)*sqrt(31)*(9 + sqrt(93))^(2/3)/24
Rapid solution
[src]
_______________ / _______________\
/ ____ | / ____ |
/ 27 3*\/ 93 | ___ / 27 3*\/ 93 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = - ---------------------- + -------------------- + I*|---------------------- + --------------------------|
_______________ 6 | _______________ 6 |
/ ____ | / ____ |
/ 27 3*\/ 93 | / 27 3*\/ 93 |
2*3 / -- + -------- |2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)$$
_______________ / _______________\
/ ____ | / ____ |
/ 27 3*\/ 93 | ___ / 27 3*\/ 93 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x2 = - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------|
_______________ 6 | _______________ 6 |
/ ____ | / ____ |
/ 27 3*\/ 93 | / 27 3*\/ 93 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)$$
_______________
/ ____
/ 27 3*\/ 93
3 / -- + --------
1 \/ 2 2
x3 = -------------------- - --------------------
_______________ 3
/ ____
/ 27 3*\/ 93
3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}$$
x3 = -(27/2 + 3*sqrt(93)/2)^(1/3)/3 + (27/2 + 3*sqrt(93)/2)^(-1/3)
Numerical answer
[src]
x1 = 0.34116390191401 + 1.16154139999725*i
x2 = 0.34116390191401 - 1.16154139999725*i
x3 = -0.682327803828019
x3 = -0.682327803828019
The graph