The coefficient of x in the quadratic equation `x^2 + px + q = 0` was taken as 17 in place of 13. its roots were found to be - 2 and - 15. Find the roots of the original equation.

Answer

To solve the problem step by step, we need to find the roots of the original quadratic equation given that the coefficient of \( x \) was incorrectly taken as 17 instead of 13. The roots corresponding to the incorrect equation were -2 and -15.

### Step 1: Identify the incorrect coefficient and roots
The incorrect quadratic equation is:
\[ x^2 + 17x + q = 0 \]
with roots \( \alpha = -2 \) and \( \beta = -15 \).

### Step 2: Use the product of the roots to find \( q \)
The product of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \alpha \beta = \frac{c}{a} \]
In our case, \( a = 1 \) (the coefficient of \( x^2 \)), and \( c = q \). Therefore:
\[ \alpha \beta = q \]
Calculating the product of the roots:
\[ \alpha \beta = (-2)(-15) = 30 \]
Thus, we find:
\[ q = 30 \]

### Step 3: Write the original quadratic equation
Now that we have the correct values for \( p \) and \( q \), we can write the original quadratic equation:
\[ x^2 + 13x + 30 = 0 \]

### Step 4: Factor the quadratic equation
To find the roots, we can factor the quadratic equation:
\[ x^2 + 13x + 30 = 0 \]
We need to find two numbers that multiply to \( 30 \) (the constant term) and add up to \( 13 \) (the coefficient of \( x \)). The numbers are \( 10 \) and \( 3 \).

Thus, we can rewrite the equation as:
\[ x^2 + 10x + 3x + 30 = 0 \]
Now, we can factor by grouping:
\[ x(x + 10) + 3(x + 10) = 0 \]
This gives us:
\[ (x + 10)(x + 3) = 0 \]

### Step 5: Solve for the roots
Setting each factor to zero gives us the roots:
\[ x + 10 = 0 \quad \Rightarrow \quad x = -10 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

### Final Answer
The roots of the original equation are:
\[ x = -10 \quad \text{and} \quad x = -3 \]

---