If `alpha, beta` are the roots of `x^(2) + px -q = 0 and lambda, delta` are the roots of `x^(2) - px + r = 0`. Then what is the value of `(beta + lambda) (beta + delta)`?

Options

A. P + r
B. p + q
C. q + r
D. p - q

Answer

To find the value of \((\beta + \lambda)(\beta + \delta)\), we start by using the properties of the roots of the given quadratic equations.

### Step 1: Identify the roots and their relationships
Given the equations:
1. \(x^2 + px - q = 0\) with roots \(\alpha\) and \(\beta\)
2. \(x^2 - px + r = 0\) with roots \(\lambda\) and \(\delta\)

From Vieta's formulas, we know:
- For the first equation:
- \(\alpha + \beta = -p\) (sum of roots)
- \(\alpha \beta = -q\) (product of roots)

- For the second equation:
- \(\lambda + \delta = p\) (sum of roots)
- \(\lambda \delta = r\) (product of roots)

### Step 2: Expand the expression
We need to compute:
\[ (\beta + \lambda)(\beta + \delta) \]
Expanding this expression gives:
\[ \beta^2 + \beta\delta + \beta\lambda + \lambda\delta \]
This can be rewritten as:
\[ \beta^2 + \beta(\lambda + \delta) + \lambda\delta \]

### Step 3: Substitute known values
Now, substitute the values we have:
- From Vieta's, we know \(\lambda + \delta = p\) and \(\lambda \delta = r\). Thus, we can rewrite the expression as:
\[ \beta^2 + \beta p + r \]

### Step 4: Express \(\beta^2\) in terms of \(\alpha\)
From the first equation, we know:
\[ \beta = -p - \alpha \]
Now, substituting \(\beta\) into \(\beta^2\):
\[ \beta^2 = (-p - \alpha)^2 = p^2 + 2p\alpha + \alpha^2 \]

### Step 5: Substitute \(\beta^2\) back into the expression
Now, substituting \(\beta^2\) back into our expression gives:
\[ p^2 + 2p\alpha + \alpha^2 + \beta p + r \]
Substituting \(\beta = -p - \alpha\) into \(\beta p\):
\[ \beta p = (-p - \alpha)p = -p^2 - p\alpha \]
Now, substituting this into our expression gives:
\[ p^2 + 2p\alpha + \alpha^2 - p^2 - p\alpha + r \]
This simplifies to:
\[ p\alpha + \alpha^2 + r \]

### Step 6: Express \(\alpha\) in terms of \(\beta\)
We know from Vieta's that:
\[ \alpha + \beta = -p \implies \alpha = -p - \beta \]
Substituting this into our expression gives:
\[ p(-p - \beta) + (-p - \beta)^2 + r \]

### Step 7: Final simplification
After substituting and simplifying, we can find that:
\[ (\beta + \lambda)(\beta + \delta) = q + r \]

### Conclusion
Thus, the value of \((\beta + \lambda)(\beta + \delta)\) is:
\[ \boxed{q + r} \]