If `alpha, beta` be the roots `x^(2)-px+q=0 and alpha', beta'` be those of `x^(2)-p'x+q'=0`, then the value of
`(alpha-alpha')^(2) +(beta-alpha')^(2) +(alpha-beta')^(2)+(beta-beta')^(2)=`…..

Answer

To solve the problem, we need to calculate the expression:

\[ (\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2 \]

where \(\alpha, \beta\) are the roots of the equation \(x^2 - px + q = 0\) and \(\alpha', \beta'\) are the roots of the equation \(x^2 - p'x + q' = 0\).

### Step 1: Identify the roots and their properties

From the first quadratic equation \(x^2 - px + q = 0\):
- The sum of the roots \(\alpha + \beta = p\)
- The product of the roots \(\alpha \beta = q\)

From the second quadratic equation \(x^2 - p'x + q' = 0\):
- The sum of the roots \(\alpha' + \beta' = p'\)
- The product of the roots \(\alpha' \beta' = q'\)

### Step 2: Expand the expression

Now we will expand the expression:

\[ (\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2 \]

Using the identity \((a - b)^2 = a^2 - 2ab + b^2\), we can expand each term:

1. \((\alpha - \alpha')^2 = \alpha^2 - 2\alpha\alpha' + \alpha'^2\)
2. \((\beta - \alpha')^2 = \beta^2 - 2\beta\alpha' + \alpha'^2\)
3. \((\alpha - \beta')^2 = \alpha^2 - 2\alpha\beta' + \beta'^2\)
4. \((\beta - \beta')^2 = \beta^2 - 2\beta\beta' + \beta'^2\)

### Step 3: Combine the expanded terms

Now, we combine all the expanded terms:

\[ = (\alpha^2 + \beta^2 + \alpha'^2 + \beta'^2) - 2(\alpha\alpha' + \beta\alpha' + \alpha\beta' + \beta\beta') \]

### Step 4: Use the identities for sums and products

We know from the properties of roots:

- \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = p^2 - 2q\)
- \(\alpha'^2 + \beta'^2 = (\alpha' + \beta')^2 - 2\alpha'\beta' = p'^2 - 2q'\)

Substituting these into our expression gives:

\[ = (p^2 - 2q) + (p'^2 - 2q') - 2(\alpha\alpha' + \beta\alpha' + \alpha\beta' + \beta\beta') \]

### Step 5: Simplify the cross terms

The cross terms can be grouped as follows:

\[ = p^2 + p'^2 - 2q - 2q' - 2(\alpha\alpha' + \beta\alpha' + \alpha\beta' + \beta\beta') \]

### Step 6: Final expression

Thus, the final expression we need to evaluate is:

\[ = p^2 + p'^2 - 2(q + q') - 2(\alpha\alpha' + \beta\alpha' + \alpha\beta' + \beta\beta') \]

### Conclusion

The value of the expression is:

\[ = p^2 + p'^2 - 2(q + q') - 2(\alpha\alpha' + \beta\alpha' + \alpha\beta' + \beta\beta') \]