If he equation `x^3+a x^2+b x+216=0` has three real roots in G.P., then `b//a` has the value equal to _____.

Explanation

To solve the problem, we need to analyze the cubic equation given and the properties of its roots. Let's go through the solution step by step.

### Step 1: Understand the roots in G.P.
Given the equation \( x^3 + ax^2 + bx + 216 = 0 \), we know that it has three real roots in geometric progression (G.P.). Let's denote the roots as \( \alpha/r, \alpha, \alpha r \), where \( \alpha \) is the middle term and \( r \) is the common ratio.

### Step 2: Use Vieta's Formulas
According to Vieta's formulas for a cubic equation \( x^3 + px^2 + qx + r = 0 \):
- The sum of the roots \( \alpha/r + \alpha + \alpha r = -a \)
- The sum of the product of the roots taken two at a time \( (\alpha/r) \alpha + \alpha (\alpha r) + (\alpha/r)(\alpha r) = b \)
- The product of the roots \( (\alpha/r) \cdot \alpha \cdot (\alpha r) = -216 \)

### Step 3: Calculate the product of the roots
From the product of the roots:
\[ \frac{\alpha^3 r}{r} = -216 \implies \alpha^3 = -216 \implies \alpha = -6 \]

### Step 4: Substitute \( \alpha \) into the sum of the roots
Now substituting \( \alpha = -6 \) into the sum of the roots:
\[ \frac{-6}{r} + (-6) + (-6r) = -a \]
This simplifies to:
\[ -\frac{6}{r} - 6 - 6r = -a \]
Multiplying through by \( -1 \):
\[ \frac{6}{r} + 6 + 6r = a \]

### Step 5: Calculate the sum of the product of the roots
Next, we calculate the sum of the product of the roots:
\[ \left(\frac{-6}{r}\right)(-6) + (-6)(-6r) + \left(\frac{-6}{r}\right)(-6r) = b \]
This simplifies to:
\[ \frac{36}{r} + 36r + 36 = b \]

### Step 6: Find \( \frac{b}{a} \)
Now we have expressions for \( a \) and \( b \):
- \( a = \frac{6}{r} + 6 + 6r \)
- \( b = \frac{36}{r} + 36r + 36 \)

To find \( \frac{b}{a} \):
\[ \frac{b}{a} = \frac{\frac{36}{r} + 36r + 36}{\frac{6}{r} + 6 + 6r} \]

### Step 7: Simplify \( \frac{b}{a} \)
Notice that both the numerator and denominator can be factored:
\[ \frac{b}{a} = \frac{6 \left( \frac{6}{r} + 6r + 6 \right)}{6 \left( \frac{1}{r} + 1 + r \right)} = 6 \]

### Final Answer
Thus, the value of \( \frac{b}{a} \) is \( 6 \).