# How do you find all solutions to #x^3+64i=0#?

Let

First let us plot the point

# " " :. x = 4(-sqrt(3)/2-1/2i) #

# " " :. x = -2sqrt(3)-2i #

# n=0 => theta = (-pi)/6 #

# " " :. x = 4(cos ((-pi)/6)+ isin ((-pi)/6)) #

# " " :. x = 4(sqrt(3)/2-1/2i) #

# " " :. x = 2sqrt(3)-2i #

# n=1 => theta = (pi)/2 #

# " " :. x = 4(cos ((pi)/2)+ isin ((pi)/2)) #

# " " :. x = 4(0+i) #

# " " :. x = 4i #

After which the pattern continues.

We can plot these solutions on the Argand Diagram

By signing up, you agree to our Terms of Service and Privacy Policy

To find all solutions to the equation (x^3 + 64i = 0), you can use the cubic formula or exploit the factorization of the equation. The given equation is in the form of a sum of cubes, so it can be factored using the sum of cubes formula:

[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) ]

Here, (a = x) and (b = 4i). Substituting into the formula:

[ x^3 + (4i)^3 = (x + 4i)(x^2 - 4ix + (4i)^2) ]

[ x^3 + 64i = (x + 4i)(x^2 - 4ix - 16) ]

Now, we set each factor equal to zero and solve for (x):

- (x + 4i = 0) gives (x = -4i).
- (x^2 - 4ix - 16 = 0) is a quadratic equation which can be solved using the quadratic formula:

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

where (a = 1), (b = -4i), and (c = -16).

[ x = \frac{4i \pm \sqrt{(-4i)^2 - 4(1)(-16)}}{2(1)} ]

[ x = \frac{4i \pm \sqrt{-16 + 64}}{2} ]

[ x = \frac{4i \pm \sqrt{48}}{2} ]

[ x = \frac{4i \pm 4\sqrt{3}i}{2} ]

[ x = 2i \pm 2\sqrt{3}i ]

So, the solutions to the equation (x^3 + 64i = 0) are (x = -4i), (x = 2i + 2\sqrt{3}i), and (x = 2i - 2\sqrt{3}i).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7