If the roots of ` 32x^3 - 48 x^2 +22 x-3=0` are in A.P then the middle root is

Options

A. 2
B. `1/2`
C. `4`
D. `1/4`

Answer

To find the middle root of the cubic equation \( 32x^3 - 48x^2 + 22x - 3 = 0 \) given that its roots are in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Understand the Roots in A.P.
Let the roots be represented as:
- \( r_1 = \alpha - d \)
- \( r_2 = \alpha \)
- \( r_3 = \alpha + d \)

Here, \( \alpha \) is the middle root and \( d \) is the common difference.

### Step 2: Use the Sum of Roots Formula
According to Vieta's formulas, the sum of the roots of the polynomial \( ax^3 + bx^2 + cx + d = 0 \) is given by:
\[ r_1 + r_2 + r_3 = -\frac{b}{a} \]
For our equation:
- \( a = 32 \)
- \( b = -48 \)

Thus, we have:
\[ (\alpha - d) + \alpha + (\alpha + d) = -\frac{-48}{32} \]

### Step 3: Simplify the Sum of Roots
This simplifies to:
\[ 3\alpha = \frac{48}{32} \]
\[ 3\alpha = \frac{3}{2} \]

### Step 4: Solve for \( \alpha \)
Now, divide both sides by 3:
\[ \alpha = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2} \]

### Step 5: Identify the Middle Root
Since \( \alpha \) is the middle root, we conclude that the middle root is:
\[ \alpha = \frac{1}{2} \]

### Final Answer
The middle root of the equation \( 32x^3 - 48x^2 + 22x - 3 = 0 \) is \( \frac{1}{2} \).

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