If sum of the roots is -2 and product of the roots is `-15`,

Options

A. 1. both the roots lie on the right-side of the V-axis
B. 2. both the roots lie on the left-side of the Z-axis
C. 3. roots lie on both the sides of thq X-axis
D. 4. one root is zero and another root is imaginary number.

Answer

To solve the problem, we need to find the quadratic equation whose roots have a sum of -2 and a product of -15. We can use the relationship between the coefficients of a quadratic equation and its roots.

### Step 1: Write the general form of a quadratic equation
The general form of a quadratic equation is:
\[ ax^2 + bx + c = 0 \]

### Step 2: Use the relationships for the sum and product of roots
For a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Given:
- Sum of the roots \( \alpha + \beta = -2 \)
- Product of the roots \( \alpha \beta = -15 \)

### Step 3: Set up the equation
Assuming \( a = 1 \) (for simplicity), we can set:
- \( -b = -2 \) which implies \( b = 2 \)
- \( c = -15 \)

Thus, the quadratic equation becomes:
\[ x^2 + 2x - 15 = 0 \]

### Step 4: Solve the quadratic equation using the quadratic formula
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting \( a = 1 \), \( b = 2 \), and \( c = -15 \):
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-15)}}{2 \cdot 1} \]

### Step 5: Calculate the discriminant
Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64 \]

### Step 6: Substitute back into the formula
Now substituting back into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{64}}{2} \]
\[ x = \frac{-2 \pm 8}{2} \]

### Step 7: Find the roots
Calculating the two possible values:
1. \( x_1 = \frac{-2 + 8}{2} = \frac{6}{2} = 3 \)
2. \( x_2 = \frac{-2 - 8}{2} = \frac{-10}{2} = -5 \)

Thus, the roots of the equation are \( 3 \) and \( -5 \).

### Conclusion
The roots \( 3 \) and \( -5 \) lie on both sides of the x-axis.