If the equation has no real root, then `lamda` lies in the interval

Options

A. `(-oo,0)`
B. `(-oo,6)`
C. `(6,oo)`
D. `(0,oo)`

Explanation

To solve the problem of determining the interval in which \( \lambda \) lies such that the equation \( x^4 - \lambda x^2 + 9 = 0 \) has no real roots, we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Equation**:
We start with the equation:
\[ x^4 - \lambda x^2 + 9 = 0 \]
To simplify, we can substitute \( x^2 = t \), transforming the equation into:
\[ t^2 - \lambda t + 9 = 0 \]

2. **Identify Coefficients**:
In the quadratic equation \( t^2 - \lambda t + 9 = 0 \), we identify:
- \( a = 1 \)
- \( b = -\lambda \)
- \( c = 9 \)

3. **Determine the Discriminant**:
For the quadratic equation to have no real roots, the discriminant must be less than zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values, we get:
\[ D = (-\lambda)^2 - 4 \cdot 1 \cdot 9 = \lambda^2 - 36 \]

4. **Set the Discriminant Condition**:
We want the discriminant to be less than zero for the equation to have no real roots:
\[ \lambda^2 - 36 < 0 \]

5. **Solve the Inequality**:
Rearranging the inequality gives:
\[ \lambda^2 < 36 \]
Taking the square root of both sides, we find:
\[ -6 < \lambda < 6 \]

6. **Conclusion**:
Therefore, the values of \( \lambda \) that ensure the equation \( x^4 - \lambda x^2 + 9 = 0 \) has no real roots lie in the interval:
\[ \lambda \in (-6, 6) \]

### Final Answer:
The interval in which \( \lambda \) lies such that the equation has no real roots is:
\[ \lambda \in (-6, 6) \]