Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation a^2*x^2-21*a^2+a*x+1=0
-
Equation 5-x=4(x-3)
-
Equation -(-y+4)+2*y+1=0
-
Equation 3^x=8
- Express {x} in terms of y where:
- -16*x-20*y=-13
- 4*x+5*y=7
- -13*x-15*y=1
- -12*x-2*y=7
- Graphing y =:
- x^3-x+1
- Factor polynomial:
- x^3-x+1
-
Identical expressions
- x^ three -x+ one = zero
- x cubed minus x plus 1 equally 0
- x to the power of three minus 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
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 1$$
Rapid solution
[src]
_______________ / _______________\
/ ____ | / ____ |
/ 27 3*\/ 69 | ___ / 27 3*\/ 69 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = ---------------------- + -------------------- + I*|- ---------------------- + --------------------------|
_______________ 6 | _______________ 6 |
/ ____ | / ____ |
/ 27 3*\/ 69 | / 27 3*\/ 69 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)$$
_______________ / _______________\
/ ____ | / ____ |
/ 27 3*\/ 69 | ___ / 27 3*\/ 69 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x2 = ---------------------- + -------------------- + I*|---------------------- - --------------------------|
_______________ 6 | _______________ 6 |
/ ____ | / ____ |
/ 27 3*\/ 69 | / 27 3*\/ 69 |
2*3 / -- + -------- |2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)$$
_______________
/ ____
/ 27 3*\/ 69
3 / -- + --------
1 \/ 2 2
x3 = - -------------------- - --------------------
_______________ 3
/ ____
/ 27 3*\/ 69
3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}$$
x3 = -(3*sqrt(69)/2 + 27/2)^(1/3)/3 - 1/(3*sqrt(69)/2 + 27/2)^(1/3)
Sum and product of roots
[src]
sum
_______________ / _______________\ _______________ / _______________\ _______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 27 3*\/ 69 | ___ / 27 3*\/ 69 | / 27 3*\/ 69 | ___ / 27 3*\/ 69 | / 27 3*\/ 69
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*\/ 69 | / 27 3*\/ 69 | / 27 3*\/ 69 | / 27 3*\/ 69 | / 27 3*\/ 69
2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- |2*3 / -- + -------- | 3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right) + \left(\left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)\right) + \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________\ | / ____ | | / ____ | | ___ / 27 3*\/ 69 | | ___ / 27 3*\/ 69 | | ___ \/ 3 *3 / -- + -------- | | ___ \/ 3 *3 / -- + -------- | | \/ 3 \/ 2 2 | | \/ 3 \/ 2 2 | I*|---------------------- - --------------------------| + I*|- ---------------------- + --------------------------| | _______________ 6 | | _______________ 6 | | / ____ | | / ____ | | / 27 3*\/ 69 | | / 27 3*\/ 69 | |2*3 / -- + -------- | | 2*3 / -- + -------- | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)$$
product
/ _______________ / _______________\\ / _______________ / _______________\\ / _______________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 27 3*\/ 69 | ___ / 27 3*\/ 69 || | / 27 3*\/ 69 | ___ / 27 3*\/ 69 || | / 27 3*\/ 69 | | 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*\/ 69 | / 27 3*\/ 69 || | / 27 3*\/ 69 | / 27 3*\/ 69 || | / 27 3*\/ 69 | |2*3 / -- + -------- | 2*3 / -- + -------- || |2*3 / -- + -------- |2*3 / -- + -------- || | 3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)\right) \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)$$
=
_________________ _______________ 2/3 _________________ 2/3 2/3 _______________ 2/3
3 / ____ 2/3 3 / ____ 3 ____ / ____\ ____ 3 / ____ 3 ___ / ____\ 3 ___ ____ / ____\ 2/3 ____ 3 / ____ 3 ___ 6 ___ ____ / ____\
9*\/ 108 + 12*\/ 69 9*2 *\/ 27 + 3*\/ 69 \/ 18 *\9 + \/ 69 / \/ 69 *\/ 108 + 12*\/ 69 \/ 2 *\27 + 3*\/ 69 / \/ 2 *\/ 69 *\27 + 3*\/ 69 / 2 *\/ 69 *\/ 27 + 3*\/ 69 \/ 2 *\/ 3 *\/ 23 *\9 + \/ 69 /
-1 + ---------------------- - ------------------------- - ---------------------- - --------------------------- + ------------------------ - ------------------------------- + ------------------------------ + ----------------------------------
32 32 8 32 8 72 32 24
$$- \frac{\sqrt[3]{18} \left(\sqrt{69} + 9\right)^{\frac{2}{3}}}{8} - \frac{\sqrt[3]{2} \sqrt{69} \left(3 \sqrt{69} + 27\right)^{\frac{2}{3}}}{72} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt[3]{3 \sqrt{69} + 27}}{32} - \frac{\sqrt{69} \sqrt[3]{12 \sqrt{69} + 108}}{32} - 1 + \frac{2^{\frac{2}{3}} \sqrt{69} \sqrt[3]{3 \sqrt{69} + 27}}{32} + \frac{9 \sqrt[3]{12 \sqrt{69} + 108}}{32} + \frac{\sqrt[3]{2} \sqrt{23} \sqrt[6]{3} \left(\sqrt{69} + 9\right)^{\frac{2}{3}}}{24} + \frac{\sqrt[3]{2} \left(3 \sqrt{69} + 27\right)^{\frac{2}{3}}}{8}$$
-1 + 9*(108 + 12*sqrt(69))^(1/3)/32 - 9*2^(2/3)*(27 + 3*sqrt(69))^(1/3)/32 - 18^(1/3)*(9 + sqrt(69))^(2/3)/8 - sqrt(69)*(108 + 12*sqrt(69))^(1/3)/32 + 2^(1/3)*(27 + 3*sqrt(69))^(2/3)/8 - 2^(1/3)*sqrt(69)*(27 + 3*sqrt(69))^(2/3)/72 + 2^(2/3)*sqrt(69)*(27 + 3*sqrt(69))^(1/3)/32 + 2^(1/3)*3^(1/6)*sqrt(23)*(9 + sqrt(69))^(2/3)/24
Numerical answer
[src]
x1 = 0.662358978622373 + 0.562279512062301*i
x2 = 0.662358978622373 - 0.562279512062301*i
x3 = -1.32471795724475
x3 = -1.32471795724475
The graph