If `b^(2)ge4ac` for the equation `ax^(4)+bx^(2)+c=0` then all the roots of the equation will be real if

Options

A. `bgt0,alt0,cgt0`
B. `blt0,agt0,cgt0`
C. `bgt0,agt0,cgt0`
D. `bgt0,alt0,clt0`

Explanation

To solve the problem, we need to analyze the condition given for the equation \( ax^4 + bx^2 + c = 0 \) and determine when all roots of this equation will be real.

### Step-by-Step Solution:

1. **Substitution**:
We start by substituting \( y = x^2 \). This transforms our equation into a quadratic equation:
\[ ay^2 + by + c = 0 \]
Here, \( y \) must be non-negative since \( y = x^2 \geq 0 \).

**Hint**: Remember that \( x^2 \) can only take non-negative values.

2. **Condition for Real Roots**:
For the quadratic equation \( ay^2 + by + c = 0 \) to have real roots, the discriminant must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Given that \( b^2 \geq 4ac \), this condition is satisfied.

**Hint**: The discriminant tells us about the nature of the roots of a quadratic equation.

3. **Roots of the Quadratic**:
Let the roots of the quadratic equation be \( y_1 \) and \( y_2 \). The sum and product of the roots can be expressed as:
\[ y_1 + y_2 = -\frac{b}{a} \quad \text{(sum of roots)} \]
\[ y_1y_2 = \frac{c}{a} \quad \text{(product of roots)} \]

**Hint**: The relationships between the coefficients and the roots are derived from Vieta's formulas.

4. **Conditions for Non-Negative Roots**:
Since \( y \) must be non-negative (as \( y = x^2 \)), both roots \( y_1 \) and \( y_2 \) must be greater than or equal to zero:
- For the sum of the roots:
\[ -\frac{b}{a} \geq 0 \implies b \text{ and } a \text{ must have opposite signs.} \]
- For the product of the roots:
\[ \frac{c}{a} \geq 0 \implies c \text{ and } a \text{ must have the same sign.} \]

**Hint**: Analyze the signs of the coefficients based on the conditions derived from the roots.

5. **Conclusion**:
- From \( b \) and \( a \) having opposite signs, we conclude that if \( a > 0 \), then \( b < 0 \), and if \( a < 0 \), then \( b > 0 \).
- From \( c \) and \( a \) having the same sign, we conclude that if \( a > 0 \), then \( c > 0 \), and if \( a < 0 \), then \( c < 0 \).

Thus, the conditions for all roots of the equation \( ax^4 + bx^2 + c = 0 \) to be real are:
- \( a > 0 \) and \( c > 0 \) with \( b < 0 \) or
- \( a < 0 \) and \( c < 0 \) with \( b > 0 \).

### Final Answer:
The roots of the equation will be real if:
- \( a > 0 \), \( c > 0 \), and \( b < 0 \) or
- \( a < 0 \), \( c < 0 \), and \( b > 0 \).