Solve the equation. You will have three solutions.
X^3-8=0
x^3-8=0
+ +8
x^3=8
2^3=8
so one of the soulution is x=2
using the difference of cubes factoring
x^2 - 8 = 0
(x-2)(x^2 + 2x + 4) = 0
x = 2
or
x = (-2 ±√-12)/2
= (-1 ± 2√3 i )/2
= -1 ± √3 i
for 3 solutions, one real, two complex
To solve the equation x^3 - 8 = 0, we can use the concept of cube roots. The equation can be rewritten as:
x^3 = 8
To find the solutions, we need to find the cube root of both sides of the equation.
Taking the cube root of both sides, we get:
x = ∛8
The cube root of 8 is 2, so one solution to the equation is x = 2.
However, remember that we need to find three solutions. In complex numbers, there are two additional cube roots of 8.
To find the other two cube roots, we can use Euler's formula. Euler's formula states that for any complex number z = a + bi, where a and b are real numbers, and i is the imaginary unit (√(-1)), the formula can be expressed as:
z = r * e^(iθ)
Here, r is the modulus or magnitude of z, and θ is the argument or phase angle of z.
In our case, z is 8, so:
z = 8 * e^(iθ)
To find the cube roots of 8, we need to find the values of θ that satisfy the equation above.
First, let's find the modulus of z:
|z| = |8 * e^(iθ)|
The modulus of 8 is 8, so:
8 = 8 * e^(iθ)
Dividing both sides by 8:
1 = e^(iθ)
Now, we need to find the values of θ that make e^(iθ) equal to 1.
Using Euler's formula, we know that e^(iθ) = cos(θ) + i * sin(θ).
So, we have:
cos(θ) + i * sin(θ) = 1
Since cos(0) = 1 and sin(0) = 0, one solution is θ = 0.
Now, we need to find the other two values of θ that make e^(iθ) equal to 1. These solutions can be found using the unit circle or by knowing the periodicity of trigonometric functions.
The values of θ that satisfy the equation cos(θ) + i * sin(θ) = 1 are:
θ = 0, 2π, and -2π
Finally, we can find the other two cube roots of 8 by setting θ to 2π and -2π:
For θ = 2π:
z = 8 * e^(i * 2π)
z = 8 * (cos(2π) + i * sin(2π))
z = 8 * (1 + 0i)
z = 8
For θ = -2π:
z = 8 * e^(i * -2π)
z = 8 * (cos(-2π) + i * sin(-2π))
z = 8 * (1 + 0i)
z = 8
So, the two additional solutions to the equation x^3 - 8 = 0 are x = 8 and x = 8.
In summary, the three solutions to the equation x^3 - 8 = 0 are x = 2, x = 8, and x = 8.