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

Answer

To find the numbers whose product is 6 and sum is -5, we can follow these steps:

### Step 1: Set up the equations
Let the two numbers be \( x \) and \( y \).
We know:
1. \( x \cdot y = 6 \) (Product)
2. \( x + y = -5 \) (Sum)

### Step 2: Express one variable in terms of the other
From the product equation, we can express \( x \) in terms of \( y \):
\[ x = \frac{6}{y} \]

### Step 3: Substitute into the sum equation
Substituting \( x \) into the sum equation gives:
\[ \frac{6}{y} + y = -5 \]

### Step 4: Clear the fraction
To eliminate the fraction, multiply the entire equation by \( y \) (assuming \( y \neq 0 \)):
\[ 6 + y^2 = -5y \]

### Step 5: Rearrange into standard quadratic form
Rearranging the equation gives:
\[ y^2 + 5y + 6 = 0 \]

### Step 6: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ y^2 + 5y + 6 = (y + 2)(y + 3) = 0 \]

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

### Step 8: Find corresponding \( x \) values
Now, we can find the corresponding \( x \) values using \( x = \frac{6}{y} \):
- If \( y = -2 \):
\[ x = \frac{6}{-2} = -3 \]
- If \( y = -3 \):
\[ x = \frac{6}{-3} = -2 \]

### Conclusion
Thus, the two numbers are \( -2 \) and \( -3 \).

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