If ` f(x) = ax^(2) + bx + c ` , where ` a ne 0, b ,c in ` R , then which of
the following conditions implies that f(x) has real roots?

Options

A. a+ b + c = 0
B. a and c are of opposite signs
C. ` 4ac - b^(2) lt 0 `
D. a and b are of opposite signs

Explanation

To determine the conditions under which the quadratic function \( f(x) = ax^2 + bx + c \) has real roots, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero:

\[ D \geq 0 \]

This means:

\[ b^2 - 4ac \geq 0 \]

Now, let's analyze the given conditions one by one to see which of them imply that \( f(x) \) has real roots.

### Step 1: Analyze the first condition \( a + b + c = 0 \)

If \( a + b + c = 0 \), then substituting \( x = 1 \) into the function gives:

\[ f(1) = a(1)^2 + b(1) + c = a + b + c = 0 \]

This means that \( x = 1 \) is a root of the equation. Since one root exists, the quadratic must have real roots. Thus, this condition implies that \( f(x) \) has real roots.

### Step 2: Analyze the second condition \( ac < 0 \)

If \( ac < 0 \), it means that \( a \) and \( c \) have opposite signs. This implies that the parabola opens upwards (if \( a > 0 \)) and crosses the x-axis (if \( c < 0 \)), or it opens downwards (if \( a < 0 \)) and also crosses the x-axis (if \( c > 0 \)).

Using the discriminant:

\[ D = b^2 - 4ac \]

Since \( ac < 0 \), \( -4ac > 0 \). Therefore, \( b^2 - 4ac > 0 \) implies that the discriminant is positive, indicating that there are two distinct real roots. Thus, this condition also implies that \( f(x) \) has real roots.

### Step 3: Analyze the third condition \( 4ac - b^2 < 0 \)

Rearranging this condition gives:

\[ b^2 - 4ac > 0 \]

This is the same condition we derived from the discriminant. Therefore, this condition also implies that \( f(x) \) has real roots.

### Step 4: Analyze the fourth condition \( ab < 0 \)

If \( ab < 0 \), it means that \( a \) and \( b \) have opposite signs. However, this condition alone does not provide information about the value of \( c \). Therefore, we cannot conclude that the quadratic has real roots based solely on this condition. Thus, this condition does not imply that \( f(x) \) has real roots.

### Conclusion

The conditions that imply that \( f(x) \) has real roots are:

1. \( a + b + c = 0 \)
2. \( ac < 0 \)
3. \( 4ac - b^2 < 0 \)

The fourth condition \( ab < 0 \) does not imply that \( f(x) \) has real roots.