Consider the following for the next two items that follow :
`2x^(2) + 3x - alpha = 0` has roots - 2 and `beta` while the equation `x^(2) - 3mx + 2m^(2) = 0` has both roots positive, where `alpha gt 0 and beta gt 0` .
If `beta.2.2m` are in GP. Then what is the value of `beta sqrt(m)`

Options

A. 1
B. 2
C. 4
D. 6

Answer

To solve the problem step by step, we will follow the given information and derive the necessary values.

### Step 1: Find the value of alpha
We are given the quadratic equation:
\[ 2x^2 + 3x - \alpha = 0 \]
with roots -2 and β. According to Vieta's formulas, the sum of the roots (−2 + β) is equal to \(-\frac{b}{a}\) and the product of the roots (−2 * β) is equal to \(\frac{c}{a}\).

From the equation:
- The sum of the roots:
\[ -2 + \beta = -\frac{3}{2} \]
This simplifies to:
\[ \beta = -\frac{3}{2} + 2 = \frac{1}{2} \]

- The product of the roots:
\[ (-2) \cdot \beta = \frac{-\alpha}{2} \]
Substituting \(\beta = \frac{1}{2}\):
\[ -2 \cdot \frac{1}{2} = \frac{-\alpha}{2} \]
This simplifies to:
\[ -1 = \frac{-\alpha}{2} \implies \alpha = 2 \]

### Step 2: Substitute alpha back into the equation
Now we have:
\[ 2x^2 + 3x - 2 = 0 \]

### Step 3: Find the roots of the equation
We will solve the quadratic equation \(2x^2 + 3x - 2 = 0\) using the factorization method:
\[ 2x^2 + 4x - x - 2 = 0 \]
Grouping the terms:
\[ (2x^2 + 4x) + (-x - 2) = 0 \]
Factoring out:
\[ 2x(x + 2) - 1(x + 2) = 0 \]
This gives:
\[ (2x - 1)(x + 2) = 0 \]
Thus, the roots are:
\[ x = \frac{1}{2} \quad \text{and} \quad x = -2 \]
So, \(\beta = \frac{1}{2}\).

### Step 4: Analyze the second quadratic equation
The second equation is:
\[ x^2 - 3mx + 2m^2 = 0 \]
We know both roots are positive. According to Vieta's formulas:
- The sum of the roots is \(3m\) (which must be positive).
- The product of the roots is \(2m^2\) (which must also be positive).

### Step 5: Determine the condition for GP
We are given that \(\beta\), 2, and \(2m\) are in geometric progression (GP). In a GP, the square of the middle term is equal to the product of the other two terms:
\[ 2^2 = \beta \cdot 2m \]
Substituting \(\beta = \frac{1}{2}\):
\[ 4 = \frac{1}{2} \cdot 2m \]
This simplifies to:
\[ 4 = m \]

### Step 6: Calculate \(\beta \sqrt{m}\)
Now we need to find \(\beta \sqrt{m}\):
\[ \beta = \frac{1}{2}, \quad m = 4 \implies \sqrt{m} = \sqrt{4} = 2 \]
Thus:
\[ \beta \sqrt{m} = \frac{1}{2} \cdot 2 = 1 \]

### Final Answer
The value of \(\beta \sqrt{m}\) is \(1\).

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