To solve the equation \( 3x^3 - 26x^2 + 52x - 24 = 0 \) with roots in Geometric Progression (G.P.), we can follow these steps:
### Step 1: Define the Roots
Let the roots of the equation be \( \frac{a}{r}, a, ar \), where \( a \) is a constant and \( r \) is the common ratio.
### Step 2: Use the Product of Roots
According to Vieta's formulas, the product of the roots of a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is given by:
\[
\text{Product of roots} = -\frac{d}{a}
\]
For our equation:
- \( a = 3 \)
- \( d = -24 \)
Thus, the product of the roots is:
\[
\frac{a}{r} \cdot a \cdot ar = a^3
\]
Setting this equal to \(-\frac{-24}{3} = 8\):
\[
a^3 = 8 \implies a = 2
\]
### Step 3: Use the Sum of Roots
The sum of the roots is given by:
\[
\frac{a}{r} + a + ar = -\frac{b}{a}
\]
For our equation:
- \( b = -26 \)
Thus, the sum of the roots is:
\[
\frac{2}{r} + 2 + 2r = \frac{26}{3}
\]
Multiplying through by \( r \) to eliminate the fraction gives:
\[
2 + 2r + 2r^2 = \frac{26}{3}r
\]
Rearranging this leads to:
\[
2r^2 - \frac{26}{3}r + 2 = 0
\]
Multiplying through by 3 to clear the fraction:
\[
6r^2 - 26r + 6 = 0
\]
### Step 4: Solve the Quadratic Equation
Now we can solve the quadratic equation \( 6r^2 - 26r + 6 = 0 \) using the quadratic formula:
\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 6, b = -26, c = 6 \):
\[
r = \frac{26 \pm \sqrt{(-26)^2 - 4 \cdot 6 \cdot 6}}{2 \cdot 6}
\]
Calculating the discriminant:
\[
r = \frac{26 \pm \sqrt{676 - 144}}{12} = \frac{26 \pm \sqrt{532}}{12} = \frac{26 \pm 2\sqrt{133}}{12} = \frac{13 \pm \sqrt{133}}{6}
\]
### Step 5: Find the Roots
Now substituting the values of \( r \) back into the roots:
1. For \( r_1 = \frac{13 + \sqrt{133}}{6} \)
2. For \( r_2 = \frac{13 - \sqrt{133}}{6} \)
The roots of the equation are:
\[
\frac{2}{r_1}, 2, 2r_1 \quad \text{and} \quad \frac{2}{r_2}, 2, 2r_2
\]
### Final Roots
Thus, the roots of the equation \( 3x^3 - 26x^2 + 52x - 24 = 0 \) are:
- \( \left( \frac{2}{\frac{13 + \sqrt{133}}{6}}, 2, 2 \cdot \frac{13 + \sqrt{133}}{6} \right) \)
- \( \left( \frac{2}{\frac{13 - \sqrt{133}}{6}}, 2, 2 \cdot \frac{13 - \sqrt{133}}{6} \right) \)
