If `alpha, beta` are the roots of equation `(k+1)x^(2)-(20k+14)x+91k+40=0,(alpha lt beta),k gt 0`, then the answer the following questions
The nature of the roots of this equation is

Options

A. imaginary
B. real and distinct
C. one real only
D. none of these

Answer

To determine the nature of the roots of the quadratic equation \((k+1)x^2 - (20k + 14)x + (91k + 40) = 0\), we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). From the given equation, we can identify:
- \(a = k + 1\)
- \(b = -(20k + 14)\)
- \(c = 91k + 40\)

### Step 2: Calculate the discriminant
The nature of the roots of a quadratic equation can be determined using the discriminant \(D\), given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-(20k + 14))^2 - 4(k + 1)(91k + 40) \]

### Step 3: Simplify the discriminant
Calculating \(b^2\):
\[ D = (20k + 14)^2 - 4(k + 1)(91k + 40) \]
Expanding \(b^2\):
\[ (20k + 14)^2 = 400k^2 + 560k + 196 \]
Now, calculate \(4ac\):
\[ 4(k + 1)(91k + 40) = 4(91k^2 + 40k + 91k + 40) = 364k^2 + 160k + 364 \]
Now substituting back into the discriminant:
\[ D = (400k^2 + 560k + 196) - (364k^2 + 160k + 364) \]

### Step 4: Combine like terms
Combining the terms:
\[ D = (400k^2 - 364k^2) + (560k - 160k) + (196 - 364) \]
This simplifies to:
\[ D = 36k^2 + 400k - 168 \]

### Step 5: Factor out common terms
Factoring out 36:
\[ D = 36(k^2 + k + 1) \]

### Step 6: Analyze the discriminant
Since \(k > 0\), the expression \(k^2 + k + 1\) is always positive for all real \(k\). Therefore, \(D > 0\).

### Conclusion
Since the discriminant \(D > 0\), the roots of the quadratic equation are real and distinct.

### Final Answer
The nature of the roots of the equation is **real and distinct**.

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