Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 258x+2x-80=700
-
Equation (2*x+9)/4-(x-2)/6=3
-
Equation x^2+2x+10=0
-
Equation -4/7*x^2+28=0
- Express {x} in terms of y where:
- 16*x+14*y=-14
- 11*x-15*y=3
- 9*x-1*y=-2
- -13*x-15*y=9
- Sum of series:
-
(x-7)^3
-
-1
- Derivative of:
-
-1
- Integral of d{x}:
-
-1
-
Identical expressions
- (x- seven)^ three =- one
- (x minus 7) cubed equally minus 1
- (x minus seven) to the power of three equally minus one
- (x-7)3=-1
- x-73=-1
- (x-7)³=-1
- (x-7) to the power of 3=-1
- x-7^3=-1
-
Similar expressions
- (x-7)^3=+1
- (x+7)^3=-1
(x-7)^3=-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\left(x - 7\right)^{3} = -1$$
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:
$$\sqrt[3]{\left(x - 7\right)^{3}} = \sqrt[3]{-1}$$
or
$$x - 7 = \sqrt[3]{-1}$$
Expand brackets in the right part
Move free summands (without x)
from left part to right part, we given:
$$x = 7 + \sqrt[3]{-1}$$
We get the answer: x = 7 + (-1)^(1/3)
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x - 7$$
then the equation will be the:
$$z^{3} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = -1$$
where
$$r = 1$$
- 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:
$$z_{1} = -1$$
$$z_{2} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$z_{3} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
do backward replacement
$$z = x - 7$$
$$x = z + 7$$
The final answer:
$$x_{1} = 6$$
$$x_{2} = \frac{15}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{3} = \frac{15}{2} + \frac{\sqrt{3} i}{2}$$
$$\left(x - 7\right)^{3} = -1$$
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:
$$\sqrt[3]{\left(x - 7\right)^{3}} = \sqrt[3]{-1}$$
or
$$x - 7 = \sqrt[3]{-1}$$
Expand brackets in the right part
-7 + x = -1^1/3
Move free summands (without x)
from left part to right part, we given:
$$x = 7 + \sqrt[3]{-1}$$
We get the answer: x = 7 + (-1)^(1/3)
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x - 7$$
then the equation will be the:
$$z^{3} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = -1$$
where
$$r = 1$$
- 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:
$$z_{1} = -1$$
$$z_{2} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$z_{3} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
do backward replacement
$$z = x - 7$$
$$x = z + 7$$
The final answer:
$$x_{1} = 6$$
$$x_{2} = \frac{15}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{3} = \frac{15}{2} + \frac{\sqrt{3} i}{2}$$
Rapid solution
[src]
x1 = 6
$$x_{1} = 6$$
___
15 I*\/ 3
x2 = -- - -------
2 2
$$x_{2} = \frac{15}{2} - \frac{\sqrt{3} i}{2}$$
___
15 I*\/ 3
x3 = -- + -------
2 2
$$x_{3} = \frac{15}{2} + \frac{\sqrt{3} i}{2}$$
x3 = 15/2 + sqrt(3)*i/2
Sum and product of roots
[src]
sum
___ ___
15 I*\/ 3 15 I*\/ 3
6 + -- - ------- + -- + -------
2 2 2 2
$$\left(6 + \left(\frac{15}{2} - \frac{\sqrt{3} i}{2}\right)\right) + \left(\frac{15}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
21
$$21$$
product
/ ___\ / ___\ |15 I*\/ 3 | |15 I*\/ 3 | 6*|-- - -------|*|-- + -------| \2 2 / \2 2 /
$$6 \left(\frac{15}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{15}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
342
$$342$$
342
Numerical answer
[src]
x1 = 7.5 - 0.866025403784439*i
x2 = 7.5 + 0.866025403784439*i
x3 = 6.0
x3 = 6.0