Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 2x^2-8=0
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Equation 8x-15=-12+5x
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Equation 5*(x+9)=-8
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Equation 7(x-1)=3x
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Identical expressions
- (one /x^ two)- one =27x- one
- (1 divide by x squared ) minus 1 equally 27x minus 1
- (one divide by x to the power of two) minus one equally 27x minus one
- (1/x2)-1=27x-1
- 1/x2-1=27x-1
- (1/x²)-1=27x-1
- (1/x to the power of 2)-1=27x-1
- 1/x^2-1=27x-1
- (1 divide by x^2)-1=27x-1
-
Similar expressions
- (1/x^2)-1=27x+1
- 27(x-19,87)=21.6
- (1/x^2)+1=27x-1
- x^2-27*x-160=0
- 27^(x-1)=1/81
- 27*x-(140)+x=28
(1/x^2)-1=27x-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$-1 + \frac{1}{x^{2}} = 27 x - 1$$
transform
$$\frac{1}{x^{3}} = 27$$
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:
$$\frac{1}{\sqrt[3]{\frac{1}{x^{3}}}} = \frac{1}{\sqrt[3]{27}}$$
or
$$x = \frac{1}{3}$$
We get the answer: x = 1/3
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$\frac{1}{z^{3}} = 27$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$\frac{e^{- 3 i p}}{r^{3}} = 27$$
where
$$r = \frac{1}{3}$$
- 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
$$p = - \frac{2 \pi N}{3}$$
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:
$$z_{1} = \frac{1}{3}$$
$$z_{2} = - \frac{1}{6} - \frac{\sqrt{3} i}{6}$$
$$z_{3} = - \frac{1}{6} + \frac{\sqrt{3} i}{6}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = \frac{1}{3}$$
$$x_{2} = - \frac{1}{6} - \frac{\sqrt{3} i}{6}$$
$$x_{3} = - \frac{1}{6} + \frac{\sqrt{3} i}{6}$$
$$-1 + \frac{1}{x^{2}} = 27 x - 1$$
transform
$$\frac{1}{x^{3}} = 27$$
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:
$$\frac{1}{\sqrt[3]{\frac{1}{x^{3}}}} = \frac{1}{\sqrt[3]{27}}$$
or
$$x = \frac{1}{3}$$
We get the answer: x = 1/3
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$\frac{1}{z^{3}} = 27$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$\frac{e^{- 3 i p}}{r^{3}} = 27$$
where
$$r = \frac{1}{3}$$
- 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
$$p = - \frac{2 \pi N}{3}$$
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:
$$z_{1} = \frac{1}{3}$$
$$z_{2} = - \frac{1}{6} - \frac{\sqrt{3} i}{6}$$
$$z_{3} = - \frac{1}{6} + \frac{\sqrt{3} i}{6}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = \frac{1}{3}$$
$$x_{2} = - \frac{1}{6} - \frac{\sqrt{3} i}{6}$$
$$x_{3} = - \frac{1}{6} + \frac{\sqrt{3} i}{6}$$
Sum and product of roots
[src]
sum
___ ___ 1 1 I*\/ 3 1 I*\/ 3 - + - - - ------- + - - + ------- 3 6 6 6 6
$$\left(\frac{1}{3} + \left(- \frac{1}{6} - \frac{\sqrt{3} i}{6}\right)\right) + \left(- \frac{1}{6} + \frac{\sqrt{3} i}{6}\right)$$
=
0
$$0$$
product
___
1 I*\/ 3
- - - ------- / ___\
6 6 | 1 I*\/ 3 |
-------------*|- - + -------|
3 \ 6 6 /
$$\frac{- \frac{1}{6} - \frac{\sqrt{3} i}{6}}{3} \left(- \frac{1}{6} + \frac{\sqrt{3} i}{6}\right)$$
=
1/27
$$\frac{1}{27}$$
1/27
Rapid solution
[src]
x1 = 1/3
$$x_{1} = \frac{1}{3}$$
___
1 I*\/ 3
x2 = - - - -------
6 6
$$x_{2} = - \frac{1}{6} - \frac{\sqrt{3} i}{6}$$
___
1 I*\/ 3
x3 = - - + -------
6 6
$$x_{3} = - \frac{1}{6} + \frac{\sqrt{3} i}{6}$$
x3 = -1/6 + sqrt(3)*i/6
Numerical answer
[src]
x1 = 0.333333333333333
x2 = -0.166666666666667 - 0.288675134594813*i
x3 = -0.166666666666667 + 0.288675134594813*i
x3 = -0.166666666666667 + 0.288675134594813*i