Factor the given expression. Then use the zero product rule and the quadratic formula to solve the problem. There should be three solutions to the equation.

x^3-2197

x=

PLEASE, help me with this one problem I am stuck on! THANKS!

To factor the expression x^3 - 2197, we can use the difference of cubes formula. The formula states that for any two numbers a and b, the expression a^3 - b^3 can be factored as (a - b)(a^2 + ab + b^2).

In this case, we have x^3 - 2197. Notice that 2197 is a perfect cube; it can be written as 13^3. So we can rewrite the expression as x^3 - 13^3.

Using the difference of cubes formula, we can factor this expression as (x - 13)(x^2 + 13x + 13^2).

Now, we can use the zero product rule to solve the equation. The zero product rule states that if the product of two factors is equal to zero, then at least one of the factors must be zero.

So we set each factor equal to zero and solve for x.

1) Set x - 13 = 0:
x = 13

2) Set x^2 + 13x + 13^2 = 0:
To solve this quadratic equation, we can use the quadratic formula. The quadratic formula states that for any quadratic equation ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 13, and c = 13^2. Plugging these values into the quadratic formula, we get:

x = (-13 ± sqrt(13^2 - 4(1)(13^2))) / (2(1))

Simplifying this expression, we have:

x = (-13 ± sqrt(169 - 4(169))) / 2
x = (-13 ± sqrt(169 - 676)) / 2
x = (-13 ± sqrt(-507)) / 2

Notice that the expression inside the square root is negative, which means there are no real solutions for this part of the equation. However, we can use complex numbers to solve it.

x = (-13 ± sqrt(-507)) / 2
x = (-13 ± sqrt(-1 * 507)) / 2
x = (-13 ± sqrt(-1) * sqrt(507)) / 2
x = (-13 ± i * sqrt(507)) / 2

Therefore, the two complex solutions are:
x = (-13 + i * sqrt(507)) / 2
x = (-13 - i * sqrt(507)) / 2

So the solutions to the equation x^3 - 2197 = 0 are:
x = 13
x = (-13 + i * sqrt(507)) / 2
x = (-13 - i * sqrt(507)) / 2