Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation log(x-3)=2
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Equation cos2x=-1
- Express {x} in terms of y where:
- 6*x-2*y=5
- -20*x+17*y=-16
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- Graphing y =:
- x^3-x-3
-
Identical expressions
- x^ three -x- three = zero
- x cubed minus x minus 3 equally 0
- x to the power of three minus x minus three equally zero
- x3-x-3=0
- x³-x-3=0
- x to the power of 3-x-3=0
- x^3-x-3=O
-
Similar expressions
- x^3-x+3=0
- x^3+x-3=0
x^3-x-3=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -3$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -3$$
Sum and product of roots
[src]
sum
_____________ / _____________ \ _____________ / _____________ \
/ _____ | / _____ | / _____ | / _____ |
/ 3 \/ 717 | ___ / 3 \/ 717 | / 3 \/ 717 | ___ / 3 \/ 717 | _____________
3 / - + ------- | \/ 3 *3 / - + ------- ___ | 3 / - + ------- |\/ 3 *3 / - + ------- ___ | / _____
\/ 2 18 1 | \/ 2 18 \/ 3 | \/ 2 18 1 | \/ 2 18 \/ 3 | / 3 \/ 717 1
- ------------------ - -------------------- + I*|- ------------------------ + --------------------| + - ------------------ - -------------------- + I*|------------------------ - --------------------| + 3 / - + ------- + --------------------
2 _____________ | 2 _____________| 2 _____________ | 2 _____________| \/ 2 18 _____________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 3 \/ 717 | / 3 \/ 717 | / 3 \/ 717 | / 3 \/ 717 | / 3 \/ 717
6*3 / - + ------- | 6*3 / - + ------- | 6*3 / - + ------- | 6*3 / - + ------- | 3*3 / - + -------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)\right)\right)$$
=
/ _____________ \ / _____________ \ | / _____ | | / _____ | | ___ / 3 \/ 717 | | ___ / 3 \/ 717 | |\/ 3 *3 / - + ------- ___ | | \/ 3 *3 / - + ------- ___ | | \/ 2 18 \/ 3 | | \/ 2 18 \/ 3 | I*|------------------------ - --------------------| + I*|- ------------------------ + --------------------| | 2 _____________| | 2 _____________| | / _____ | | / _____ | | / 3 \/ 717 | | / 3 \/ 717 | | 6*3 / - + ------- | | 6*3 / - + ------- | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right) + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)$$
product
/ _____________ / _____________ \\ / _____________ / _____________ \\ | / _____ | / _____ || | / _____ | / _____ || | / 3 \/ 717 | ___ / 3 \/ 717 || | / 3 \/ 717 | ___ / 3 \/ 717 || / _____________ \ | 3 / - + ------- | \/ 3 *3 / - + ------- ___ || | 3 / - + ------- |\/ 3 *3 / - + ------- ___ || | / _____ | | \/ 2 18 1 | \/ 2 18 \/ 3 || | \/ 2 18 1 | \/ 2 18 \/ 3 || | / 3 \/ 717 1 | |- ------------------ - -------------------- + I*|- ------------------------ + --------------------||*|- ------------------ - -------------------- + I*|------------------------ - --------------------||*|3 / - + ------- + --------------------| | 2 _____________ | 2 _____________|| | 2 _____________ | 2 _____________|| |\/ 2 18 _____________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 3 \/ 717 | / 3 \/ 717 || | / 3 \/ 717 | / 3 \/ 717 || | / 3 \/ 717 | | 6*3 / - + ------- | 6*3 / - + ------- || | 6*3 / - + ------- | 6*3 / - + ------- || | 3*3 / - + ------- | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right)\right) \left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}\right)$$
=
3
$$3$$
3
Rapid solution
[src]
_____________ / _____________ \
/ _____ | / _____ |
/ 3 \/ 717 | ___ / 3 \/ 717 |
3 / - + ------- | \/ 3 *3 / - + ------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x1 = - ------------------ - -------------------- + I*|- ------------------------ + --------------------|
2 _____________ | 2 _____________|
/ _____ | / _____ |
/ 3 \/ 717 | / 3 \/ 717 |
6*3 / - + ------- | 6*3 / - + ------- |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right)$$
_____________ / _____________ \
/ _____ | / _____ |
/ 3 \/ 717 | ___ / 3 \/ 717 |
3 / - + ------- |\/ 3 *3 / - + ------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x2 = - ------------------ - -------------------- + I*|------------------------ - --------------------|
2 _____________ | 2 _____________|
/ _____ | / _____ |
/ 3 \/ 717 | / 3 \/ 717 |
6*3 / - + ------- | 6*3 / - + ------- |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)$$
_____________
/ _____
/ 3 \/ 717 1
x3 = 3 / - + ------- + --------------------
\/ 2 18 _____________
/ _____
/ 3 \/ 717
3*3 / - + -------
\/ 2 18
$$x_{3} = \frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}$$
x3 = 1/(3*(sqrt(717)/18 + 3/2)^(1/3)) + (sqrt(717)/18 + 3/2)^(1/3)
Numerical answer
[src]
x1 = -0.83584994082858 + 1.04686931884998*i
x2 = 1.67169988165716
x3 = -0.83584994082858 - 1.04686931884998*i
x3 = -0.83584994082858 - 1.04686931884998*i
The graph