Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^3+x-3=0
-
Equation 5+2x=1
-
Equation -7x+4=2x-14
-
Equation x^3-2*x^2-15*x=0
- Express {x} in terms of y where:
- 14*x-9*y=-15
- -8*x+19*y=-15
- -20*x+10*y=19
- -3*x-5*y=0
- Factor polynomial:
- x^3+x-3
- Derivative of:
- x^3+x-3
-
Identical expressions
- x^ three +x- three = zero
- x cubed plus x minus 3 equally 0
- x to the power of three plus 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 \/ 741 | ___ / 3 \/ 741 | / 3 \/ 741 | ___ / 3 \/ 741 | _____________
3 / - + ------- | \/ 3 *3 / - + ------- ___ | 3 / - + ------- |\/ 3 *3 / - + ------- ___ | / _____
\/ 2 18 1 | \/ 2 18 \/ 3 | \/ 2 18 1 | \/ 2 18 \/ 3 | / 3 \/ 741 1
- ------------------ + -------------------- + I*|- ------------------------ - --------------------| + - ------------------ + -------------------- + I*|------------------------ + --------------------| + 3 / - + ------- - --------------------
2 _____________ | 2 _____________| 2 _____________ | 2 _____________| \/ 2 18 _____________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 3 \/ 741 | / 3 \/ 741 | / 3 \/ 741 | / 3 \/ 741 | / 3 \/ 741
6*3 / - + ------- | 6*3 / - + ------- | 6*3 / - + ------- | 6*3 / - + ------- | 3*3 / - + -------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(- \frac{1}{3 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2}\right)\right)\right)$$
=
/ _____________ \ / _____________ \ | / _____ | | / _____ | | ___ / 3 \/ 741 | | ___ / 3 \/ 741 | |\/ 3 *3 / - + ------- ___ | | \/ 3 *3 / - + ------- ___ | | \/ 2 18 \/ 3 | | \/ 2 18 \/ 3 | I*|------------------------ + --------------------| + I*|- ------------------------ - --------------------| | 2 _____________| | 2 _____________| | / _____ | | / _____ | | / 3 \/ 741 | | / 3 \/ 741 | | 6*3 / - + ------- | | 6*3 / - + ------- | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}\right) + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2}\right)$$
product
/ _____________ / _____________ \\ / _____________ / _____________ \\ | / _____ | / _____ || | / _____ | / _____ || | / 3 \/ 741 | ___ / 3 \/ 741 || | / 3 \/ 741 | ___ / 3 \/ 741 || / _____________ \ | 3 / - + ------- | \/ 3 *3 / - + ------- ___ || | 3 / - + ------- |\/ 3 *3 / - + ------- ___ || | / _____ | | \/ 2 18 1 | \/ 2 18 \/ 3 || | \/ 2 18 1 | \/ 2 18 \/ 3 || | / 3 \/ 741 1 | |- ------------------ + -------------------- + I*|- ------------------------ - --------------------||*|- ------------------ + -------------------- + I*|------------------------ + --------------------||*|3 / - + ------- - --------------------| | 2 _____________ | 2 _____________|| | 2 _____________ | 2 _____________|| |\/ 2 18 _____________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 3 \/ 741 | / 3 \/ 741 || | / 3 \/ 741 | / 3 \/ 741 || | / 3 \/ 741 | | 6*3 / - + ------- | 6*3 / - + ------- || | 6*3 / - + ------- | 6*3 / - + ------- || | 3*3 / - + ------- | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}\right)\right) \left(- \frac{1}{3 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}\right)$$
=
3
$$3$$
3
Rapid solution
[src]
_____________ / _____________ \
/ _____ | / _____ |
/ 3 \/ 741 | ___ / 3 \/ 741 |
3 / - + ------- | \/ 3 *3 / - + ------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x1 = - ------------------ + -------------------- + I*|- ------------------------ - --------------------|
2 _____________ | 2 _____________|
/ _____ | / _____ |
/ 3 \/ 741 | / 3 \/ 741 |
6*3 / - + ------- | 6*3 / - + ------- |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}\right)$$
_____________ / _____________ \
/ _____ | / _____ |
/ 3 \/ 741 | ___ / 3 \/ 741 |
3 / - + ------- |\/ 3 *3 / - + ------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x2 = - ------------------ + -------------------- + I*|------------------------ + --------------------|
2 _____________ | 2 _____________|
/ _____ | / _____ |
/ 3 \/ 741 | / 3 \/ 741 |
6*3 / - + ------- | 6*3 / - + ------- |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}}{2}\right)$$
_____________
/ _____
/ 3 \/ 741 1
x3 = 3 / - + ------- - --------------------
\/ 2 18 _____________
/ _____
/ 3 \/ 741
3*3 / - + -------
\/ 2 18
$$x_{3} = - \frac{1}{3 \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}} + \sqrt[3]{\frac{3}{2} + \frac{\sqrt{741}}{18}}$$
x3 = -1/(3*(3/2 + sqrt(741)/18)^(1/3)) + (3/2 + sqrt(741)/18)^(1/3)
Numerical answer
[src]
x1 = 1.21341166276223
x2 = -0.606705831381115 - 1.45061224918844*i
x3 = -0.606705831381115 + 1.45061224918844*i
x3 = -0.606705831381115 + 1.45061224918844*i
The graph