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Answer :
To solve the cubic equation [tex]\( 4g^3 - 23g^2 - 151 = 0 \)[/tex], we can break down the process into several steps. Cubic equations can have complex solutions, so using algebraic and numerical methods, we find the roots of the equation. Here's how to approach solving a cubic polynomial like this.
### Step-by-Step Solution
1. Identify the cubic equation:
We start with the equation:
[tex]\[
4g^3 - 23g^2 - 151 = 0
\][/tex]
2. Simplify if possible:
Our given equation is already in its simplest form. We now focus on finding its roots.
3. Use the general approach for solving cubic equations:
A cubic equation [tex]\( ax^3 + bx^2 + cx + d = 0 \)[/tex] has solutions that can be found using various methods, including factoring, the Rational Root Theorem, or numerical solutions.
4. Apply the appropriate cubic solution formula:
For the given equation:
[tex]\[
4g^3 - 23g^2 - 151 = 0
\][/tex]
The solutions are generally more complex and may involve cube roots and imaginary numbers. For simplicity, let's note down the solutions derived using an algebraic solver.
### Roots of the Equation:
From solving the equation, we get three complex roots (including possible real parts):
1. [tex]\( g_1 = \frac{23}{12} + \left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}} + \frac{529}{144\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}}} \)[/tex]
2. [tex]\( g_2 = \frac{23}{12} + \frac{529}{144 \left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}}} + \left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}} \)[/tex]
3. [tex]\( g_3 = \frac{529}{144 \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}}} + \frac{23}{12} + \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}} \)[/tex]
These solutions showcase the typical nature of cubic roots which includes both real and complex numbers. This method guarantees the identification of the correct roots without making manual computational mistakes, given the complexity of such a polynomial.
By listing out the complex roots with their respective combinations, we have fully solved the cubic equation and identified all the possible solutions for [tex]\( g \)[/tex].
### Step-by-Step Solution
1. Identify the cubic equation:
We start with the equation:
[tex]\[
4g^3 - 23g^2 - 151 = 0
\][/tex]
2. Simplify if possible:
Our given equation is already in its simplest form. We now focus on finding its roots.
3. Use the general approach for solving cubic equations:
A cubic equation [tex]\( ax^3 + bx^2 + cx + d = 0 \)[/tex] has solutions that can be found using various methods, including factoring, the Rational Root Theorem, or numerical solutions.
4. Apply the appropriate cubic solution formula:
For the given equation:
[tex]\[
4g^3 - 23g^2 - 151 = 0
\][/tex]
The solutions are generally more complex and may involve cube roots and imaginary numbers. For simplicity, let's note down the solutions derived using an algebraic solver.
### Roots of the Equation:
From solving the equation, we get three complex roots (including possible real parts):
1. [tex]\( g_1 = \frac{23}{12} + \left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}} + \frac{529}{144\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right) \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}}} \)[/tex]
2. [tex]\( g_2 = \frac{23}{12} + \frac{529}{144 \left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}}} + \left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}} \)[/tex]
3. [tex]\( g_3 = \frac{529}{144 \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}}} + \frac{23}{12} + \left(\frac{5\sqrt{515967}}{144} + \frac{44783}{1728}\right)^{\frac{1}{3}} \)[/tex]
These solutions showcase the typical nature of cubic roots which includes both real and complex numbers. This method guarantees the identification of the correct roots without making manual computational mistakes, given the complexity of such a polynomial.
By listing out the complex roots with their respective combinations, we have fully solved the cubic equation and identified all the possible solutions for [tex]\( g \)[/tex].
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