If p,q are the roots of the equation `x^(2)+px+q=0` where both p and q are non-zero, then (p,q)=

Options

A. (1,2)
B. (1,-2)
C. (-1,2)
D. (-1,-2)

Answer

To solve the problem where \( p \) and \( q \) are the roots of the quadratic equation \( x^2 + px + q = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = p \)
- \( c = q \)

### Step 2: Use the relationships of roots
According to Vieta's formulas:
- The sum of the roots \( p + q = -\frac{b}{a} = -p \)
- The product of the roots \( pq = \frac{c}{a} = q \)

### Step 3: Set up the equations
From the sum of the roots, we have:
\[ p + q = -p \]
Rearranging gives:
\[ p + q + p = 0 \implies 2p + q = 0 \implies q = -2p \]

From the product of the roots, we have:
\[ pq = q \]
Since \( q \) is non-zero, we can divide both sides by \( q \):
\[ p = 1 \]

### Step 4: Substitute \( p \) back to find \( q \)
Now substituting \( p = 1 \) into the equation \( q = -2p \):
\[ q = -2(1) = -2 \]

### Conclusion
Thus, the values of \( p \) and \( q \) are:
\[ (p, q) = (1, -2) \]