If `a,` `b `, `c in R` and `3b^(2)-8ac lt 0`, then the equation `ax^(4)+bx^(3)+cx^(2)+5x-7=0` has

Options

A. all real roots
B. all imaginary roots
C. exactly two real and two imaginary roots
D. none

Explanation

To solve the problem, we need to analyze the given polynomial equation \( f(x) = ax^4 + bx^3 + cx^2 + 5x - 7 = 0 \) under the condition \( 3b^2 - 8ac < 0 \).

### Step-by-Step Solution:

1. **Define the Function**:
We start with the polynomial function:
\[ f(x) = ax^4 + bx^3 + cx^2 + 5x - 7 \]

2. **First Derivative**:
We find the first derivative of \( f(x) \):
\[ f'(x) = 4ax^3 + 3bx^2 + 2cx + 5 \]

3. **Second Derivative**:
Next, we find the second derivative of \( f(x) \):
\[ f''(x) = 12ax^2 + 6bx + 2c \]

4. **Critical Points**:
To find the critical points, we set the first derivative equal to zero:
\[ f'(x) = 0 \implies 4ax^3 + 3bx^2 + 2cx + 5 = 0 \]
The number of real roots of this cubic equation will determine the behavior of \( f(x) \).

5. **Discriminant of the Derivative**:
We analyze the nature of the roots of \( f'(x) \) using the discriminant. The discriminant \( D \) of a cubic equation \( Ax^3 + Bx^2 + Cx + D = 0 \) is given by:
\[ D = 18ABCD - 4B^3D + B^2C^2 - 4AC^2 - 27A^2D^2 \]
However, we can simplify our analysis using the condition given in the problem.

6. **Using the Condition**:
From the problem, we know:
\[ 3b^2 - 8ac < 0 \]
This implies that the discriminant of the second derivative \( f''(x) \) is negative, leading to the conclusion that \( f'(x) = 0 \) has only one real root.

7. **Conclusion on Roots of \( f(x) \)**:
Since \( f'(x) \) has only one real root, \( f(x) \) will change its monotonicity at this point. Therefore, \( f(x) \) must have exactly two real roots.

Thus, we conclude that the equation \( ax^4 + bx^3 + cx^2 + 5x - 7 = 0 \) has exactly two real roots.

### Final Answer:
The equation has **exactly two real roots**.

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