If the roots of ` x^3 -14x^2 + 56x-64 =0` are in G.P then the middle root is

Options

A. 4
B. `-4`
C. `6`
D. 8

Answer

To find the middle root of the equation \( x^3 - 14x^2 + 56x - 64 = 0 \) given that the roots are in geometric progression (G.P.), we can follow these steps:

### Step 1: Identify the roots in G.P.
Let the roots be \( a/t, a, at \), where \( a \) is the middle root and \( t \) is the common ratio of the G.P.

### Step 2: Use Vieta's formulas
According to Vieta's formulas for a cubic equation \( x^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( (a/t + a + at) \) is equal to \( -b \).
- The sum of the products of the roots taken two at a time \( (a/t \cdot a + a \cdot at + at \cdot a/t) \) is equal to \( c \).
- The product of the roots \( (a/t \cdot a \cdot at) \) is equal to \( -d \).

### Step 3: Apply Vieta's formulas to our equation
From the equation \( x^3 - 14x^2 + 56x - 64 = 0 \):
- The sum of the roots \( a/t + a + at = 14 \).
- The sum of the products of the roots \( a^2(t + 1 + 1/t) = 56 \).
- The product of the roots \( a^3 = 64 \).

### Step 4: Calculate the product of the roots
From the product of the roots:
\[ a^3 = 64 \]
Taking the cube root:
\[ a = \sqrt[3]{64} = 4 \]

### Step 5: Conclusion
The middle root \( a \) is \( 4 \).

### Final Answer
The middle root is \( 4 \).

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