Alright, let's solve the equation [tex]\(5 - 3x^3 = 86\)[/tex] step-by-step.
1. Start with the given equation:
[tex]\[
5 - 3x^3 = 86
\][/tex]
2. Isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
-3x^3 = 86 - 5
\][/tex]
Simplify the right-hand side:
[tex]\[
-3x^3 = 81
\][/tex]
3. Divide both sides by -3:
[tex]\[
x^3 = \frac{81}{-3}
\][/tex]
Simplify the division:
[tex]\[
x^3 = -27
\][/tex]
4. Take the cube root of both sides to solve for [tex]\(x\)[/tex]:
To solve [tex]\(x^3 = -27\)[/tex], we take the cube root of both sides. Note that the cube root of [tex]\(-27\)[/tex] has three complex roots.
The three roots are:
[tex]\[
x = -3, \quad x = \frac{3}{2} - \frac{3\sqrt{3}i}{2}, \quad x = \frac{3}{2} + \frac{3\sqrt{3}i}{2}
\][/tex]
So, the solutions to the equation [tex]\(5 - 3x^3 = 86\)[/tex] are:
[tex]\[
x = -3, \quad x = \frac{3}{2} - \frac{3\sqrt{3}i}{2}, \quad x = \frac{3}{2} + \frac{3\sqrt{3}i}{2}
\][/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the given equation.
