Find the numbers whose : (i) product `= 6` and sum `=5`

Answer

To solve the problem of finding two numbers whose product is 6 and sum is 5, we can follow these steps:

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we know:
- The product of the numbers:
\[ xy = 6 \]
- The sum of the numbers:
\[ x + y = 5 \]

### Step 3: Express One Variable in Terms of the Other
From the sum equation, we can express \( x \) in terms of \( y \):
\[ x = 5 - y \]

### Step 4: Substitute into the Product Equation
Now, substitute \( x \) in the product equation:
\[ (5 - y)y = 6 \]

### Step 5: Expand and Rearrange the Equation
Expanding the equation gives:
\[ 5y - y^2 = 6 \]
Rearranging it to standard quadratic form:
\[ y^2 - 5y + 6 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we need to factor the quadratic equation \( y^2 - 5y + 6 \):
We look for two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of \( y \)):
The factors are:
\[ (y - 2)(y - 3) = 0 \]

### Step 7: Solve for \( y \)
Setting each factor to zero gives us the possible values for \( y \):
1. \( y - 2 = 0 \) → \( y = 2 \)
2. \( y - 3 = 0 \) → \( y = 3 \)

### Step 8: Find Corresponding Values of \( x \)
Now, we can find the corresponding values of \( x \) for each \( y \):
- If \( y = 2 \):
\[ x = 5 - 2 = 3 \]
- If \( y = 3 \):
\[ x = 5 - 3 = 2 \]

### Conclusion
Thus, the two numbers are:
\[ \boxed{2 \text{ and } 3} \]