The coefficient of `x` in a quadratic equation `x^2 + px +q=0` was taken as 17 in place of 13 and its roots found to be `-3` and `-10`. The roots of the original equation are

Options

A. 2,15
B. 10, 3
C. -10, -3
D. -2, -15

Answer

To find the roots of the original quadratic equation \(x^2 + px + q = 0\), we will follow these steps:

### Step 1: Understand the given information
The coefficient of \(x\) was mistakenly taken as 17 instead of 13. The roots of the equation with the incorrect coefficient are given as \(-3\) and \(-10\).

### Step 2: Use the relationship between roots and coefficients
For a quadratic equation of the form \(x^2 + px + q = 0\):
- The sum of the roots (denoted as \(S\)) is given by \(-p\).
- The product of the roots (denoted as \(P\)) is given by \(q\).

Given the roots \(-3\) and \(-10\):
- The sum of the roots \(S = -3 + (-10) = -13\).
- The product of the roots \(P = (-3) \times (-10) = 30\).

### Step 3: Relate the sum and product to the coefficients
From the sum of the roots, we have:
\[ -p = -13 \implies p = 13 \]
From the product of the roots, we have:
\[ q = 30 \]

### Step 4: Write the original quadratic equation
Now that we have \(p\) and \(q\), we can write the original quadratic equation:
\[ x^2 + 13x + 30 = 0 \]

### Step 5: Factor the quadratic equation
We need to factor the equation \(x^2 + 13x + 30\). We look for two numbers that multiply to \(30\) and add to \(13\):
- The numbers \(3\) and \(10\) satisfy this condition.

Thus, we can factor the equation as:
\[ (x + 3)(x + 10) = 0 \]

### Step 6: Find the roots of the original equation
Setting each factor to zero gives us the roots:
1. \(x + 3 = 0 \implies x = -3\)
2. \(x + 10 = 0 \implies x = -10\)

### Conclusion
The roots of the original quadratic equation \(x^2 + 13x + 30 = 0\) are \(-3\) and \(-10\).