Solve each of the following equatins :
`x^(2)-(p+q)x+pq=0`

Answer

To solve the quadratic equation \( x^2 - (p + q)x + pq = 0 \), we will use the factorization method. Here are the steps:

### Step 1: Write the equation
The given equation is:
\[ x^2 - (p + q)x + pq = 0 \]

### Step 2: Identify the coefficients
In the quadratic equation of the form \( ax^2 + bx + c = 0 \):
- \( a = 1 \)
- \( b = -(p + q) \)
- \( c = pq \)

### Step 3: Factor the quadratic expression
We need to factor the expression \( x^2 - (p + q)x + pq \). We look for two numbers that multiply to \( pq \) (the constant term) and add up to \( -(p + q) \) (the coefficient of \( x \)).

The two numbers that satisfy these conditions are \( -p \) and \( -q \):
- \( (-p) + (-q) = -(p + q) \)
- \( (-p) \cdot (-q) = pq \)

Thus, we can factor the quadratic as:
\[ (x - p)(x - q) = 0 \]

### Step 4: Set each factor to zero
Now we set each factor equal to zero:
1. \( x - p = 0 \) ⟹ \( x = p \)
2. \( x - q = 0 \) ⟹ \( x = q \)

### Step 5: Write the final solution
The solutions (roots) of the equation \( x^2 - (p + q)x + pq = 0 \) are:
\[ x = p \quad \text{or} \quad x = q \]