Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation tan(x)=0
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Equation x^2-2x-24=0
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Equation k^2-4*k+3=0
- Equation x³+18=0
- Express {x} in terms of y where:
- 10*x+9*y=-4
- -15*x-19*y=2
- -11*x-14*y=-19
- -17*x-15*y=-17
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Identical expressions
- x³+ eighteen = zero
- x³ plus 18 equally 0
- x³ plus eighteen equally zero
- x³+18=O
-
Similar expressions
- x³-18=0
x³+18=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{3} + 18 = 0$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{x^{3}} = \sqrt[3]{-18}$$
or
$$x = \sqrt[3]{-18}$$
Expand brackets in the right part
We get the answer: x = (-18)^(1/3)
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{3} = -18$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = -18$$
where
$$r = \sqrt[3]{18}$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = -1$$
so
$$\cos{\left(3 p \right)} = -1$$
and
$$\sin{\left(3 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{3} + \frac{\pi}{3}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = \frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
$$z_{2} = - \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}$$
$$z_{3} = - \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = \frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
$$x_{2} = - \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}$$
$$x_{3} = - \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
$$x^{3} + 18 = 0$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{x^{3}} = \sqrt[3]{-18}$$
or
$$x = \sqrt[3]{-18}$$
Expand brackets in the right part
x = -18^1/3
We get the answer: x = (-18)^(1/3)
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{3} = -18$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = -18$$
where
$$r = \sqrt[3]{18}$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = -1$$
so
$$\cos{\left(3 p \right)} = -1$$
and
$$\sin{\left(3 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{3} + \frac{\pi}{3}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = \frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
$$z_{2} = - \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}$$
$$z_{3} = - \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = \frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
$$x_{2} = - \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}$$
$$x_{3} = - \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 18$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 18$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 18$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 18$$
Sum and product of roots
[src]
sum
3 ____ 3 ___ 2/3 3 ____ 3 ___ 6 ___ 3 ____ 3 ___ 2/3 3 ___ 6 ___
\/ 18 3*\/ 2 *3 \/ 18 3*I*\/ 2 *\/ 3 \/ 18 3*\/ 2 *3 3*I*\/ 2 *\/ 3
- ------ - ------------ + ------ + --------------- + - ------ + ------------ - ---------------
4 4 2 2 4 4 2
$$\left(- \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}\right) + \left(\left(- \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}\right) + \left(\frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}\right)\right)$$
=
0
$$0$$
product
/ 3 ____ 3 ___ 2/3\ /3 ____ 3 ___ 6 ___\ / 3 ____ 3 ___ 2/3 3 ___ 6 ___\ | \/ 18 3*\/ 2 *3 | |\/ 18 3*I*\/ 2 *\/ 3 | | \/ 18 3*\/ 2 *3 3*I*\/ 2 *\/ 3 | |- ------ - ------------|*|------ + ---------------|*|- ------ + ------------ - ---------------| \ 4 4 / \ 2 2 / \ 4 4 2 /
$$\left(\frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}\right) \left(- \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}\right) \left(- \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}\right)$$
=
-18
$$-18$$
-18
Rapid solution
[src]
3 ____ 3 ___ 2/3
\/ 18 3*\/ 2 *3
x1 = - ------ - ------------
4 4
$$x_{1} = - \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{\sqrt[3]{18}}{4}$$
3 ____ 3 ___ 6 ___
\/ 18 3*I*\/ 2 *\/ 3
x2 = ------ + ---------------
2 2
$$x_{2} = \frac{\sqrt[3]{18}}{2} + \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
3 ____ 3 ___ 2/3 3 ___ 6 ___
\/ 18 3*\/ 2 *3 3*I*\/ 2 *\/ 3
x3 = - ------ + ------------ - ---------------
4 4 2
$$x_{3} = - \frac{\sqrt[3]{18}}{4} + \frac{3 \sqrt[3]{2} \cdot 3^{\frac{2}{3}}}{4} - \frac{3 \sqrt[3]{2} \sqrt[6]{3} i}{2}$$
x3 = -18^(1/3)/4 + 3*2^(1/3)*3^(2/3)/4 - 3*2^(1/3)*3^(1/6)*i/2
Numerical answer
[src]
x1 = 1.31037069710445 - 2.26962862413435*i
x2 = 1.31037069710445 + 2.26962862413435*i
x3 = -2.6207413942089
x3 = -2.6207413942089