If the roots of the equation `4 beta^(2) + lambda beta - 2 = 0` are of the from `(k)/(k+1) and (k+1)/(k+2)` then what is the value of `lambda` ?

Options

A. 2k
B. 7
C. 2
D. K+1

Answer

To find the value of \(\lambda\) for the quadratic equation \(4\beta^2 + \lambda \beta - 2 = 0\) given that its roots are of the form \(\frac{k}{k+1}\) and \(\frac{k+1}{k+2}\), we can follow these steps:

### Step 1: Identify the roots
The roots of the equation are given as:
\[ \alpha = \frac{k}{k+1}, \quad \beta = \frac{k+1}{k+2} \]

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation of the form \(ax^2 + bx + c = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

Here, \(a = 4\), \(b = \lambda\), and \(c = -2\).

### Step 3: Calculate the sum of the roots
First, we calculate the sum of the roots:
\[ \alpha + \beta = \frac{k}{k+1} + \frac{k+1}{k+2} \]
To add these fractions, we find a common denominator:
\[ \alpha + \beta = \frac{k(k+2) + (k+1)(k+1)}{(k+1)(k+2)} \]
Expanding the numerator:
\[ = \frac{k^2 + 2k + k^2 + 2k + 1}{(k+1)(k+2)} = \frac{2k^2 + 4k + 1}{(k+1)(k+2)} \]

According to Vieta's, we have:
\[ \frac{2k^2 + 4k + 1}{(k+1)(k+2)} = -\frac{\lambda}{4} \]

### Step 4: Calculate the product of the roots
Next, we calculate the product of the roots:
\[ \alpha \beta = \frac{k}{k+1} \cdot \frac{k+1}{k+2} = \frac{k}{k+2} \]
According to Vieta's, we have:
\[ \frac{k}{k+2} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 5: Solve for \(k\)
From the equation:
\[ \frac{k}{k+2} = -\frac{1}{2} \]
Cross-multiplying gives:
\[ 2k = -k - 2 \implies 3k = -2 \implies k = -\frac{2}{3} \]

### Step 6: Substitute \(k\) back to find \(\lambda\)
Now substitute \(k = -\frac{2}{3}\) back into the sum of the roots equation:
\[ \alpha + \beta = \frac{2\left(-\frac{2}{3}\right)^2 + 4\left(-\frac{2}{3}\right) + 1}{\left(-\frac{2}{3}+1\right)\left(-\frac{2}{3}+2\right)} \]
Calculating the numerator:
\[ = \frac{2 \cdot \frac{4}{9} - \frac{8}{3} + 1}{\left(\frac{1}{3}\right)\left(\frac{4}{3}\right)} = \frac{\frac{8}{9} - \frac{24}{9} + \frac{9}{9}}{\frac{4}{9}} = \frac{-7/9}{4/9} = -\frac{7}{4} \]
Setting this equal to \(-\frac{\lambda}{4}\):
\[ -\frac{7}{4} = -\frac{\lambda}{4} \implies \lambda = 7 \]

### Final Answer
Thus, the value of \(\lambda\) is:
\[ \boxed{7} \]