If sum of the roots is `-11` and product of the roots is 24,

Options

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

Answer

To solve the problem step by step, we need to form a quadratic equation using the given sum and product of the roots.

### Step 1: Form the Quadratic Equation
Given:
- Sum of the roots (α + β) = -11
- Product of the roots (α * β) = 24

We can use the standard form of a quadratic equation:
\[ x^2 - (sum \, of \, roots)x + (product \, of \, roots) = 0 \]

Substituting the values we have:
\[ x^2 - (-11)x + 24 = 0 \]
This simplifies to:
\[ x^2 + 11x + 24 = 0 \]

### Step 2: Identify Coefficients
From the equation \( x^2 + 11x + 24 = 0 \), we identify:
- a = 1
- b = 11
- c = 24

### Step 3: Use the Quadratic Formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of a, b, and c:
\[ x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \]

### Step 4: Calculate the Discriminant
Calculate \( b^2 - 4ac \):
\[ 11^2 - 4 \cdot 1 \cdot 24 = 121 - 96 = 25 \]

### Step 5: Substitute Back into the Formula
Now substitute the discriminant back into the formula:
\[ x = \frac{-11 \pm \sqrt{25}}{2} \]
\[ x = \frac{-11 \pm 5}{2} \]

### Step 6: Calculate the Roots
Now we calculate the two possible values for x:
1. For the positive root:
\[ x_1 = \frac{-11 + 5}{2} = \frac{-6}{2} = -3 \]

2. For the negative root:
\[ x_2 = \frac{-11 - 5}{2} = \frac{-16}{2} = -8 \]

### Conclusion
The roots of the equation are:
\[ \alpha = -3 \]
\[ \beta = -8 \]