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 find the roots of the original quadratic equation \(x^2 + px + q = 0\), we start by analyzing the information given in the problem.

1. **Identify the modified equation**: The coefficient of \(x\) was mistakenly taken as 17 instead of 13. Therefore, the modified quadratic equation is:
\[ x^2 + 17x + q = 0 \]

2. **Use the roots provided**: The roots of this modified equation are given as -2 and -15. We can use Vieta's formulas, which state that for a quadratic equation \(ax^2 + bx + c = 0\):
- The sum of the roots \(r_1 + r_2 = -\frac{b}{a}\)
- The product of the roots \(r_1 \cdot r_2 = \frac{c}{a}\)

Here, \(r_1 = -2\) and \(r_2 = -15\).

3. **Calculate the sum of the roots**:
\[ r_1 + r_2 = -2 + (-15) = -17 \]
According to Vieta's formulas:
\[ -\frac{17}{1} = -17 \]
This confirms that the coefficient of \(x\) in the modified equation is correct.

4. **Calculate the product of the roots**:
\[ r_1 \cdot r_2 = (-2) \cdot (-15) = 30 \]
According to Vieta's formulas:
\[ \frac{q}{1} = 30 \implies q = 30 \]

5. **Formulate the modified equation**: Now we have the modified equation:
\[ x^2 + 17x + 30 = 0 \]

6. **Find the original equation**: The original equation has \(p = 13\) instead of 17. Therefore, the original quadratic equation is:
\[ x^2 + 13x + 30 = 0 \]

7. **Find the roots of the original equation**: We can find the roots of the original equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 13\), and \(c = 30\):
\[ x = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 1 \cdot 30}}{2 \cdot 1} \]
\[ = \frac{-13 \pm \sqrt{169 - 120}}{2} \]
\[ = \frac{-13 \pm \sqrt{49}}{2} \]
\[ = \frac{-13 \pm 7}{2} \]

Now, calculating the two possible values:
- For the positive case:
\[ x = \frac{-13 + 7}{2} = \frac{-6}{2} = -3 \]
- For the negative case:
\[ x = \frac{-13 - 7}{2} = \frac{-20}{2} = -10 \]

8. **Final roots**: The roots of the original equation \(x^2 + 13x + 30 = 0\) are:
\[ x = -3 \quad \text{and} \quad x = -10 \]