Question

Find the exact solutions of \(x^{3}+2=0\).

Short answer

The exact solution is \( x = -\root{3}{2} \).

Step-by-step solution

Isolate the variable term

Start by subtracting 2 from both sides of the equation to isolate the variable term:\[ x^3 + 2 = 0 \] becomes\[ x^3 = -2 \]

Take the cube root of both sides

To solve for \( x \), take the cube root of both sides of the equation:\[ x = \root{3}{-2} \]

Simplify the cube root of a negative number

The cube root of -2 is -\( \root{3}{2} \):\[ x = -\root{3}{2} \]

Key concepts

Understanding Cube Roots

Cube roots are essential when dealing with cubic equations. To find the cube root of a number is to determine which number, when multiplied by itself twice (three times in total), gives the original number.
For instance, the cube root of 8 is 2, because 2 * 2 * 2 = 8. Similarly, the cube root of -8 is -2 because (-2) * (-2) * (-2) = -8.
Cube roots can handle negative numbers because multiplying three negative numbers results in a negative product.
Unlike square roots, which always have two solutions (one positive and one negative), cube roots always have one real solution.
Knowing this helps in solving cubic equations like the one in our exercise.

Isolating the Variable

Isolating the variable is often the first step in solving an equation. The goal is to get the variable term by itself on one side of the equation.
In our exercise, we started with the equation: \(x^3 + 2 = 0\)
To isolate the variable term, we subtracted 2 from both sides to obtain: \(x^3 = -2\)
Isolating the variable simplifies our job, making it easier to solve for the exact value of the variable.

Finding Exact Solutions

Finding the exact solution of an equation means expressing the solution in its simplest form without approximations.
In our exercise, once we isolated the variable term to get: \(x^3 = -2\)
We took the cube root of both sides to solve for \(x\): \(x = \root{3}{-2}\)
Finally, because the cube root of -2 is -\( \root{3}{2}\), we reached the exact solution: \(x = -\root{3}{2}\)
Exact solutions are beneficial because they offer a precise and clear understanding of the answer, eliminating ambiguity.