Sure, let's solve the equation [tex]\(6x^3 - 23x + 20 = 0\)[/tex] step-by-step.
### Step 1: Identify the type of equation
The given equation is a cubic polynomial of the form [tex]\(ax^3 + bx^2 + cx + d = 0\)[/tex]. In this case, [tex]\(a = 6\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = -23\)[/tex], and [tex]\(d = 20\)[/tex].
### Step 2: Standard form of the equation
Rewrite the equation for clarity:
[tex]\[ 6x^3 - 23x + 20 = 0 \][/tex]
### Step 3: Finding the roots
To solve a cubic equation of the form [tex]\(ax^3 + bx^2 + cx + d = 0\)[/tex], we often use methods such as factoring, synthetic division, or the cubic formula. However, this particular cubic polynomial doesn’t lend itself easily to simple factoring or synthetic division due to the specific coefficients and the lack of a quadratic term.
### Step 4: Solving the cubic equation
Given the complexity of the polynomial, solving it exactly usually requires finding the roots using more advanced algebraic methods which involve manipulation into a depressed cubic form, trigonometric solutions, or leveraging numerical methods.
The roots for the equation [tex]\(6x^3 - 23x + 20 = 0\)[/tex] are complex and are detailed as follows:
### Step 5: Writing the solutions
The three roots of the cubic equation [tex]\(6x^3 - 23x + 20 = 0\)[/tex] are:
[tex]\[
x_1 = -\frac{23}{6(-\frac{1}{2} - \frac{\sqrt{3}i}{2})(\frac{\sqrt{8066}}{4} + 45)^{1/3}} - \frac{(-\frac{1}{2} - \frac{\sqrt{3}i}{2})(\frac{\sqrt{8066}}{4} + 45)^{1/3}}{3}
\][/tex]
[tex]\[
x_2 = -\frac{(-\frac{1}{2} + \frac{\sqrt{3}i}{2})(\frac{\sqrt{8066}}{4} + 45)^{1/3}}{3} - \frac{23}{6(-\frac{1}{2} + \frac{\sqrt{3}i}{2})(\frac{\sqrt{8066}}{4} + 45)^{1/3}}
\][/tex]
[tex]\[
x_3 = -\frac{(\frac{\sqrt{8066}}{4} + 45)^{1/3}}{3} - \frac{23}{6(\frac{\sqrt{8066}}{4} + 45)^{1/3}}
\][/tex]
The solutions are indicative of the roots involving both real and complex numbers due to the terms with imaginary unit [tex]\(i\)[/tex].
### Conclusion
The cubic equation [tex]\(6x^3 - 23x + 20 = 0\)[/tex] has three roots which are expressed in a somewhat complex form, involving cube roots and imaginary numbers. These indicate that the solutions are not simple rational numbers and require a symbolic form to be properly represented.
