The quadratic equations `Sigma ((x-q)(x-r))/((p-q)(p-r))-1=0` or `Sigma ((x-q)(x-r))/((p-q)(p-r))p^(2)-x^(2)=0` have

Options

A. two solutions
B. one solution
C. no solution
D. infinite solutions

Answer

To solve the given quadratic equations, we will analyze both equations step by step.

### Step 1: Analyze the first equation
The first equation is given as:

\[ \Sigma \frac{(x-q)(x-r)}{(p-q)(p-r)} - 1 = 0 \]

This can be rewritten as:

\[ \Sigma \frac{(x-q)(x-r)}{(p-q)(p-r)} = 1 \]

### Step 2: Expand the summation
The summation involves cyclic permutations of \(p\), \(q\), and \(r\). We can express this as:

\[ \frac{(x-q)(x-r)}{(p-q)(p-r)} + \frac{(x-p)(x-r)}{(q-p)(q-r)} + \frac{(x-p)(x-q)}{(r-p)(r-q)} = 1 \]

### Step 3: Substitute values
Now, we will substitute \(x = p\), \(x = q\), and \(x = r\) into the equation to check for roots.

1. **Substituting \(x = p\)**:
\[ f(p) = \frac{(p-q)(p-r)}{(p-q)(p-r)} + 0 + 0 = 1 - 1 = 0 \]

2. **Substituting \(x = q\)**:
\[ f(q) = 0 + \frac{(q-p)(q-r)}{(q-p)(q-r)} + 0 = 1 - 1 = 0 \]

3. **Substituting \(x = r\)**:
\[ f(r) = 0 + 0 + \frac{(r-p)(r-q)}{(r-p)(r-q)} = 1 - 1 = 0 \]

### Step 4: Conclusion for the first equation
Since \(f(p) = 0\), \(f(q) = 0\), and \(f(r) = 0\), we have three roots for a quadratic equation. A quadratic equation cannot have more than two distinct roots unless it is an identity. Therefore, this implies that the equation is an identity, and thus it has infinite solutions.

### Step 5: Analyze the second equation
The second equation is given as:

\[ \Sigma \frac{(x-q)(x-r)}{(p-q)(p-r)}p^2 - x^2 = 0 \]

### Step 6: Expand the second equation
Similar to the first equation, we can express this as:

\[ \frac{(x-q)(x-r)}{(p-q)(p-r)}p^2 + \frac{(x-p)(x-r)}{(q-p)(q-r)}q^2 + \frac{(x-p)(x-q)}{(r-p)(r-q)}r^2 - x^2 = 0 \]

### Step 7: Substitute values for the second equation
We will again substitute \(x = p\), \(x = q\), and \(x = r\):

1. **Substituting \(x = p\)**:
\[ f(p) = p^2 - p^2 = 0 \]

2. **Substituting \(x = q\)**:
\[ f(q) = q^2 - q^2 = 0 \]

3. **Substituting \(x = r\)**:
\[ f(r) = r^2 - r^2 = 0 \]

### Step 8: Conclusion for the second equation
Similar to the first equation, since \(f(p) = 0\), \(f(q) = 0\), and \(f(r) = 0\), we conclude that this equation also has infinite solutions.

### Final Conclusion
Both quadratic equations have infinite solutions.