Ramesh and Mahesh solve an equation. In solving Ramesh commits a mistake in constant term and find the roots are 8 and 2. Mahesh commits a mistake in the coefficient of x and find the roots -9 and -1. The corret roots are

Options

A. `-8,2`
B. 9,1
C. 9,-1
D. `-8,-2`

Answer

To find the correct roots of the quadratic equation that Ramesh and Mahesh were solving, we can follow these steps:

### Step 1: Identify the roots found by Ramesh and Mahesh
Ramesh found the roots to be 8 and 2, while Mahesh found the roots to be -9 and -1.

### Step 2: Calculate the sum and product of the roots found by Ramesh
- The sum of the roots (Ramesh) = 8 + 2 = 10
- The product of the roots (Ramesh) = 8 * 2 = 16

### Step 3: Calculate the sum and product of the roots found by Mahesh
- The sum of the roots (Mahesh) = -9 + (-1) = -10
- The product of the roots (Mahesh) = -9 * -1 = 9

### Step 4: Set up the equations based on Ramesh's and Mahesh's findings
From Ramesh's findings, we can write the quadratic equation as:
\[ x^2 - (sum \ of \ roots) \cdot x + (product \ of \ roots) = 0 \]
So, it becomes:
\[ x^2 - 10x + 16 = 0 \]

From Mahesh's findings, we can write the quadratic equation as:
\[ x^2 - (sum \ of \ roots) \cdot x + (product \ of \ roots) = 0 \]
So, it becomes:
\[ x^2 + 10x + 9 = 0 \]

### Step 5: Find the correct sum and product of the roots
The correct sum of the roots should be the average of the sums found by Ramesh and Mahesh:
- Correct sum = (10 + (-10)) / 2 = 0

The correct product of the roots should be the average of the products found by Ramesh and Mahesh:
- Correct product = (16 + 9) / 2 = 12.5

### Step 6: Write the correct quadratic equation
Now we can write the correct quadratic equation:
\[ x^2 - (correct \ sum) \cdot x + (correct \ product) = 0 \]
This gives us:
\[ x^2 + 0 \cdot x + 12.5 = 0 \]
or simply:
\[ x^2 + 12.5 = 0 \]

### Step 7: Solve for the roots
To solve for the roots:
\[ x^2 = -12.5 \]
Taking the square root of both sides:
\[ x = \pm i\sqrt{12.5} \]
This means the roots are complex numbers.

### Final Answer
The correct roots are \( x = \pm i\sqrt{12.5} \).

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