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 first identify if it follows any known factorization patterns. In this case, the expression follows the pattern of a difference of cubes:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Using this pattern, we can factor x^3 - 2197 as (x - 13)(x^2 + 13x + 169).
Now, to solve the equation (x - 13)(x^2 + 13x + 169) = 0, we can apply the zero product rule. According to the zero product rule, 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. x - 13 = 0
x = 13
2. x^2 + 13x + 169 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = 13, and c = 169. Substituting these values into the quadratic formula, we get:
x = (-13 ± √(13^2 - 4(1)(169))) / (2(1))
Simplifying further:
x = (-13 ± √(169 - 676)) / 2
x = (-13 ± √(-507)) / 2
Since we have a negative value under the square root, we know that there are no real solutions. However, we can simplify the expression:
x = (-13 ± √(169) * √(-3)) / 2
x = (-13 ± 13i√3) / 2
Therefore, the three solutions to the equation x^3 - 2197 = 0 are:
x = 13
x = (-13 + 13i√3) / 2
x = (-13 - 13i√3) / 2