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

Answer

To find the 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 \( a \) and \( b \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The product of the numbers: \( ab = -6 \)
2. The sum of the numbers: \( a + b = -5 \)

### Step 3: Express One Variable in Terms of the Other
From the product equation, we can express \( a \) in terms of \( b \):
\[ a = \frac{-6}{b} \]

### Step 4: Substitute into the Sum Equation
Substituting \( a \) into the sum equation:
\[ \frac{-6}{b} + b = -5 \]

### Step 5: Clear the Fraction
To eliminate the fraction, multiply through by \( b \) (assuming \( b \neq 0 \)):
\[ -6 + b^2 = -5b \]

### Step 6: Rearrange the Equation
Rearranging gives us:
\[ b^2 + 5b - 6 = 0 \]

### Step 7: Factor the Quadratic Equation
Now, we can factor the quadratic:
\[ b^2 + 6b - b - 6 = 0 \]
Grouping the terms:
\[ (b + 6)(b - 1) = 0 \]

### Step 8: Solve for \( b \)
Setting each factor to zero gives us:
1. \( b + 6 = 0 \) → \( b = -6 \)
2. \( b - 1 = 0 \) → \( b = 1 \)

### Step 9: Find Corresponding Values of \( a \)
Now, we can find \( a \) for each value of \( b \):
1. If \( b = -6 \):
\[ a = \frac{-6}{-6} = 1 \]
2. If \( b = 1 \):
\[ a = \frac{-6}{1} = -6 \]

### Conclusion
Thus, the two numbers are \( 1 \) and \( -6 \).

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