The values of P for which the difference between the roots of the equation `x^(2)-px+10=0` is 3 are

Options

A. `+-3`
B. `+-5`
C. `+-6`
D. `+-7`

Answer

To solve the problem, we need to find the values of \( P \) for which the difference between the roots of the quadratic equation \( x^2 - px + 10 = 0 \) is 3.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \( x^2 - px + 10 = 0 \). Here, we can identify:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -p \) (coefficient of \( x \))
- \( c = 10 \) (constant term)

2. **Use the relationships of roots**:
Let the roots of the equation be \( \alpha \) and \( \beta \). From Vieta's formulas, we know:
- The sum of the roots: \( \alpha + \beta = -\frac{b}{a} = p \)
- The product of the roots: \( \alpha \beta = \frac{c}{a} = 10 \)

3. **Set up the equation for the difference of the roots**:
We are given that the difference between the roots is 3:
\[ \alpha - \beta = 3 \]

4. **Express \( \alpha \) and \( \beta \)**:
From the equations above, we can express \( \alpha \) and \( \beta \) in terms of \( p \):
- \( \alpha + \beta = p \)
- \( \alpha - \beta = 3 \)

We can solve these two equations simultaneously. Adding the two equations:
\[ (\alpha + \beta) + (\alpha - \beta) = p + 3 \implies 2\alpha = p + 3 \implies \alpha = \frac{p + 3}{2} \]
Subtracting the second equation from the first:
\[ (\alpha + \beta) - (\alpha - \beta) = p - 3 \implies 2\beta = p - 3 \implies \beta = \frac{p - 3}{2} \]

5. **Substitute into the product of roots**:
Now, we substitute \( \alpha \) and \( \beta \) into the product equation:
\[ \alpha \beta = 10 \]
Substituting the expressions for \( \alpha \) and \( \beta \):
\[ \left(\frac{p + 3}{2}\right) \left(\frac{p - 3}{2}\right) = 10 \]
Simplifying this gives:
\[ \frac{(p + 3)(p - 3)}{4} = 10 \]
\[ (p + 3)(p - 3) = 40 \]
\[ p^2 - 9 = 40 \]
\[ p^2 = 49 \]

6. **Solve for \( p \)**:
Taking the square root of both sides:
\[ p = \pm 7 \]

### Conclusion:
The values of \( P \) for which the difference between the roots of the equation \( x^2 - px + 10 = 0 \) is 3 are:
\[ p = 7 \quad \text{and} \quad p = -7 \]