Solve the following equation ,given that the root of each are in A.P .
(i) `8x^3-36x^2-18x+81=0`

Answer

To solve the equation \(8x^3 - 36x^2 - 18x + 81 = 0\) given that the roots are in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is:
\[ 8x^3 - 36x^2 - 18x + 81 = 0 \]
We can identify the coefficients as follows:
- \(a = 8\)
- \(b = -36\)
- \(c = -18\)
- \(d = 81\)

### Step 2: Express the roots in terms of A.P.
Let the roots be \(a - d\), \(a\), and \(a + d\), where \(a\) is the middle term and \(d\) is the common difference.

### Step 3: Use the sum of the roots
According to Vieta's formulas, the sum of the roots is given by:
\[ \text{Sum of roots} = -\frac{b}{a} = -\frac{-36}{8} = \frac{36}{8} = \frac{9}{2} \]
Thus, we have:
\[ (a - d) + a + (a + d) = 3a = \frac{9}{2} \]
From this, we can solve for \(a\):
\[ 3a = \frac{9}{2} \implies a = \frac{9}{6} = \frac{3}{2} \]

### Step 4: Use the product of the roots
The product of the roots is given by:
\[ \text{Product of roots} = -\frac{d}{a} = -\frac{81}{8} \]
Thus, we have:
\[ (a - d) \cdot a \cdot (a + d) = -\frac{81}{8} \]
Using the identity \( (a - d)(a + d) = a^2 - d^2 \), we can rewrite the product:
\[ (a^2 - d^2) \cdot a = -\frac{81}{8} \]
Substituting \(a = \frac{3}{2}\):
\[ \left(\left(\frac{3}{2}\right)^2 - d^2\right) \cdot \frac{3}{2} = -\frac{81}{8} \]
Calculating \(a^2\):
\[ \left(\frac{9}{4} - d^2\right) \cdot \frac{3}{2} = -\frac{81}{8} \]
Multiplying both sides by \(8\):
\[ 8 \cdot \left(\frac{9}{4} - d^2\right) \cdot \frac{3}{2} = -81 \]
This simplifies to:
\[ 12 \cdot \left(\frac{9}{4} - d^2\right) = -81 \]
Dividing by 12:
\[ \frac{9}{4} - d^2 = -\frac{27}{4} \]
Rearranging gives:
\[ \frac{9}{4} + \frac{27}{4} = d^2 \implies \frac{36}{4} = d^2 \implies d^2 = 9 \implies d = \pm 3 \]

### Step 5: Find the roots
Now substituting back to find the roots:
1. \(a - d = \frac{3}{2} - 3 = -\frac{3}{2}\)
2. \(a = \frac{3}{2}\)
3. \(a + d = \frac{3}{2} + 3 = \frac{9}{2}\)

Thus, the roots are:
\[ -\frac{3}{2}, \frac{3}{2}, \frac{9}{2} \]

### Final Answer
The roots of the equation \(8x^3 - 36x^2 - 18x + 81 = 0\) are:
\[ -\frac{3}{2}, \frac{3}{2}, \frac{9}{2} \]