The coefficient of x in the equation `x^2+px+q=0` was wrongly written as 17 in place of 13 and the roots thus found were -2 and -15. The roots of the correct equation are (A) `15.-2` (B) `-3,-10` (C) `-13,30` (D) `4,13`

Options

A. `-1`
B. `-3`
C. `-5`
D. `-10`

Answer

To solve the problem step by step, we will analyze the given information and derive the correct quadratic equation.

### Step 1: Identify the incorrect quadratic equation
The problem states that the coefficient of \(x\) was written as 17 instead of 13. Therefore, the incorrect quadratic equation is:
\[ x^2 + 17x + q = 0 \]

### Step 2: Use the roots to find \(q\)
The roots of the incorrect equation are given as -2 and -15. According to Vieta's formulas, the sum of the roots (\( \alpha + \beta \)) is equal to -p, and the product of the roots (\( \alpha \cdot \beta \)) is equal to \(q\).

Calculating the product of the roots:
\[ \alpha \cdot \beta = (-2) \cdot (-15) = 30 \]
Thus, we have:
\[ q = 30 \]

### Step 3: Write the correct quadratic equation
Now that we have \(q\), we can write the correct quadratic equation with the correct coefficient of \(x\):
\[ x^2 + 13x + 30 = 0 \]

### Step 4: Find the roots of the correct quadratic equation
To find the roots of the correct equation \(x^2 + 13x + 30 = 0\), we can use the factorization method. We need to find two numbers that multiply to 30 and add up to 13.

The numbers that satisfy this condition are 3 and 10. Thus, we can factor the equation as:
\[ (x + 3)(x + 10) = 0 \]

Setting each factor to zero gives us the roots:
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x + 10 = 0 \quad \Rightarrow \quad x = -10 \]

### Conclusion
The roots of the correct quadratic equation \(x^2 + 13x + 30 = 0\) are:
\[ \text{Roots: } -3 \text{ and } -10 \]

### Final Answer
The correct option is (B) \(-3, -10\).