To solve the equation [tex]\(6x^3 - 3 - 7x = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that satisfy the equation. This is a cubic equation, so it will have up to three solutions, which may be real or complex. Let's solve it step-by-step.
1. Rewrite the equation in standard form:
[tex]\[
6x^3 - 7x - 3 = 0
\][/tex]
2. Identify the polynomial components:
The cubic polynomial is [tex]\(6x^3 - 7x - 3\)[/tex].
3. Find the roots (solutions) of the cubic polynomial:
The solutions of the cubic equation [tex]\(6x^3 - 7x - 3 = 0\)[/tex] can be found using algebraic methods for solving cubic equations or by utilizing the general cubic formula. The complete derivation and algebraic manipulation are quite involved and typically require advanced methods such as Cardano's formula.
After solving the cubic equation, we find the following solutions for [tex]\(x\)[/tex]:
[tex]\[
\left( -\frac{1}{2} - \frac{\sqrt{3}i}{2} \right) \left( \frac{\sqrt{43}}{108} + \frac{1}{4} \right)^{1/3} + \frac{7}{18 \left( -\frac{1}{2} - \frac{\sqrt{3}i}{2} \right) \left( \frac{\sqrt{43}}{108} + \frac{1}{4} \right)^{1/3}}
\][/tex]
[tex]\[
\frac{7}{18 \left( -\frac{1}{2} + \frac{\sqrt{3}i}{2} \right) \left( \frac{\sqrt{43}}{108} + \frac{1}{4} \right)^{1/3}} + \left( -\frac{1}{2} + \frac{\sqrt{3}i}{2} \right) \left( \frac{\sqrt{43}}{108} + \frac{1}{4} \right)^{1/3}
\][/tex]
[tex]\[
\frac{7}{18 \left( \frac{\sqrt{43}}{108} + \frac{1}{4} \right)^{1/3}} + \left( \frac{\sqrt{43}}{108} + \frac{1}{4} \right)^{1/3}
\][/tex]
These solutions are expressed in terms of complex numbers and roots. The expressions might appear daunting, but they are precise solutions derived using methods for solving cubic equations.
