If the equation `x^(2) - bx + 1 = 0` does not proses real roots. Then which one of the following is correct ?

Options

A. `-3 lt b lt 3`
B. `-2 lt b lt 2`
C. `b lt 2`
D. `b lt - 2`

Answer

To determine the conditions under which the quadratic equation \( x^2 - bx + 1 = 0 \) does not have real roots, we need to analyze the discriminant of the equation.

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**:
The given quadratic equation is:
\[ x^2 - bx + 1 = 0 \]

2. **Recall the Discriminant Condition**:
For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
The equation does not have real roots if the discriminant is less than zero:
\[ D < 0 \]

3. **Substitute the Coefficients**:
In our equation, \( a = 1 \), \( b = -b \), and \( c = 1 \). Thus, the discriminant becomes:
\[ D = (-b)^2 - 4(1)(1) = b^2 - 4 \]

4. **Set Up the Inequality**:
For the equation to not have real roots, we need:
\[ b^2 - 4 < 0 \]

5. **Solve the Inequality**:
Rearranging the inequality gives:
\[ b^2 < 4 \]
Taking the square root of both sides (and considering both positive and negative roots), we find:
\[ -2 < b < 2 \]

6. **Conclusion**:
Therefore, the condition for the quadratic equation \( x^2 - bx + 1 = 0 \) to not have real roots is:
\[ b \text{ is between } -2 \text{ and } 2 \]

### Final Answer:
The correct option is:
- \( b \) is greater than -2 and smaller than 2.

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