If the equation `x^(2)-x+ m^(2) = 0` has no real roots then m can satisfy

Options

A. m=0
B. `m lt (-1)/(2)`
C. `m lt (1)/(2)`
D. `m in R`

Answer

To determine the values of \( m \) for which the equation \( x^2 - x + m^2 = 0 \) has no real roots, we will analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, \( a = 1 \), \( b = -1 \), and \( c = m^2 \).

2. **Write the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-1)^2 - 4 \cdot 1 \cdot m^2 = 1 - 4m^2 \]

3. **Set the condition for no real roots**:
For the quadratic equation to have no real roots, the discriminant must be less than zero:
\[ D < 0 \implies 1 - 4m^2 < 0 \]

4. **Solve the inequality**:
Rearranging the inequality:
\[ 1 < 4m^2 \]
Dividing both sides by 4:
\[ \frac{1}{4} < m^2 \]

5. **Taking the square root**:
Taking the square root of both sides gives:
\[ |m| > \frac{1}{2} \]
This implies:
\[ m < -\frac{1}{2} \quad \text{or} \quad m > \frac{1}{2} \]

6. **Conclusion**:
Therefore, the values of \( m \) for which the equation \( x^2 - x + m^2 = 0 \) has no real roots are:
\[ m < -\frac{1}{2} \quad \text{or} \quad m > \frac{1}{2} \]

### Final Answer:
The correct option is \( m < -\frac{1}{2} \) or \( m > \frac{1}{2} \).