The equal `x^(2)-px+q=0, p,q, in R` has no real roots if :

Options

A. `p^(2) gt 4p`
B. `p^(2) lt 4q`
C. `p^(2)=4q`
D. none of these

Answer

To determine the conditions under which the quadratic equation \( x^2 - px + q = 0 \) has no real roots, we can analyze the discriminant of the equation. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

In our case, the coefficients are:
- \( a = 1 \)
- \( b = -p \)
- \( c = q \)

Thus, the discriminant for our equation becomes:

\[ D = (-p)^2 - 4(1)(q) = p^2 - 4q \]

For the quadratic equation to have no real roots, the discriminant must be less than zero:

\[ D < 0 \]

This leads us to the inequality:

\[ p^2 - 4q < 0 \]

Rearranging this inequality gives us:

\[ p^2 < 4q \]

Therefore, the condition for the equation \( x^2 - px + q = 0 \) to have no real roots is:

\[ p^2 < 4q \]

### Step-by-Step Solution:
1. **Identify the quadratic equation**: The given equation is \( x^2 - px + q = 0 \).
2. **Write down the discriminant**: The discriminant \( D \) is calculated as \( D = p^2 - 4q \).
3. **Set the condition for no real roots**: For the quadratic to have no real roots, we need \( D < 0 \).
4. **Formulate the inequality**: This gives us \( p^2 - 4q < 0 \).
5. **Rearrange the inequality**: Rearranging leads to \( p^2 < 4q \).

### Conclusion:
The correct condition under which the quadratic equation \( x^2 - px + q = 0 \) has no real roots is:

**Answer**: \( p^2 < 4q \)