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- Equation:
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Identical expressions
- (x+ nine)^ three =- twenty-seven
- (x plus 9) cubed equally minus 27
- (x plus nine) to the power of three equally minus twenty minus seven
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- x+9^3=-27
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Similar expressions
- (x+9)^3=+27
- (x-9)^3=-27
(x+9)^3=-27 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\left(x + 9\right)^{3} = -27$$
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:
$$\sqrt[3]{\left(x + 9\right)^{3}} = \sqrt[3]{-27}$$
or
$$x + 9 = 3 \sqrt[3]{-1}$$
Expand brackets in the right part
Move free summands (without x)
from left part to right part, we given:
$$x = -9 + 3 \sqrt[3]{-1}$$
We get the answer: x = -9 + 3*(-1)^(1/3)
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x + 9$$
then the equation will be the:
$$z^{3} = -27$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = -27$$
where
$$r = 3$$
- 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
$$p = \frac{2 \pi N}{3} + \frac{\pi}{3}$$
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:
$$z_{1} = -3$$
$$z_{2} = \frac{3}{2} - \frac{3 \sqrt{3} i}{2}$$
$$z_{3} = \frac{3}{2} + \frac{3 \sqrt{3} i}{2}$$
do backward replacement
$$z = x + 9$$
$$x = z - 9$$
The final answer:
$$x_{1} = -12$$
$$x_{2} = - \frac{15}{2} - \frac{3 \sqrt{3} i}{2}$$
$$x_{3} = - \frac{15}{2} + \frac{3 \sqrt{3} i}{2}$$
$$\left(x + 9\right)^{3} = -27$$
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:
$$\sqrt[3]{\left(x + 9\right)^{3}} = \sqrt[3]{-27}$$
or
$$x + 9 = 3 \sqrt[3]{-1}$$
Expand brackets in the right part
9 + x = -3*1^1/3
Move free summands (without x)
from left part to right part, we given:
$$x = -9 + 3 \sqrt[3]{-1}$$
We get the answer: x = -9 + 3*(-1)^(1/3)
All other 2 root(s) is the complex numbers.
do replacement:
$$z = x + 9$$
then the equation will be the:
$$z^{3} = -27$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = -27$$
where
$$r = 3$$
- 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
$$p = \frac{2 \pi N}{3} + \frac{\pi}{3}$$
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:
$$z_{1} = -3$$
$$z_{2} = \frac{3}{2} - \frac{3 \sqrt{3} i}{2}$$
$$z_{3} = \frac{3}{2} + \frac{3 \sqrt{3} i}{2}$$
do backward replacement
$$z = x + 9$$
$$x = z - 9$$
The final answer:
$$x_{1} = -12$$
$$x_{2} = - \frac{15}{2} - \frac{3 \sqrt{3} i}{2}$$
$$x_{3} = - \frac{15}{2} + \frac{3 \sqrt{3} i}{2}$$
Sum and product of roots
[src]
sum
___ ___
15 3*I*\/ 3 15 3*I*\/ 3
-12 + - -- - --------- + - -- + ---------
2 2 2 2
$$\left(-12 + \left(- \frac{15}{2} - \frac{3 \sqrt{3} i}{2}\right)\right) + \left(- \frac{15}{2} + \frac{3 \sqrt{3} i}{2}\right)$$
=
-27
$$-27$$
product
/ ___\ / ___\
| 15 3*I*\/ 3 | | 15 3*I*\/ 3 |
-12*|- -- - ---------|*|- -- + ---------|
\ 2 2 / \ 2 2 /
$$- 12 \left(- \frac{15}{2} - \frac{3 \sqrt{3} i}{2}\right) \left(- \frac{15}{2} + \frac{3 \sqrt{3} i}{2}\right)$$
=
-756
$$-756$$
-756
Rapid solution
[src]
x1 = -12
$$x_{1} = -12$$
___
15 3*I*\/ 3
x2 = - -- - ---------
2 2
$$x_{2} = - \frac{15}{2} - \frac{3 \sqrt{3} i}{2}$$
___
15 3*I*\/ 3
x3 = - -- + ---------
2 2
$$x_{3} = - \frac{15}{2} + \frac{3 \sqrt{3} i}{2}$$
x3 = -15/2 + 3*sqrt(3)*i/2
Numerical answer
[src]
x1 = -7.5 + 2.59807621135332*i
x2 = -12.0
x3 = -7.5 - 2.59807621135332*i
x3 = -7.5 - 2.59807621135332*i