For a quadratic inequation `ax^2+bx+cgt0`, and `agt0` if you have double roots, the values of x that satisfy the given inequation

Options

A. `-oolexleoo`
B. `R-{alpha,beta}`
C. `-ooltxltoo`
D. all the real values except one value

Answer

To solve the quadratic inequality \( ax^2 + bx + c > 0 \) given that \( a > 0 \) and the equation has double roots, we can follow these steps:

### Step 1: Understand the condition of double roots
For a quadratic equation \( ax^2 + bx + c = 0 \) to have double roots, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Since we have double roots, we set:
\[ D = 0 \implies b^2 - 4ac = 0 \]

### Step 2: Identify the root
When the discriminant is zero, the quadratic equation has exactly one root, which can be calculated using the formula:
\[ x = \frac{-b}{2a} \]
This root is the point where the quadratic touches the x-axis.

### Step 3: Analyze the quadratic function
Since \( a > 0 \), the parabola opens upwards. This means that the quadratic function \( ax^2 + bx + c \) will be greater than zero for all values of \( x \) except at the double root where it equals zero.

### Step 4: Determine the solution set
The values of \( x \) that satisfy the inequality \( ax^2 + bx + c > 0 \) will be all real numbers except the double root:
\[ x \in (-\infty, \frac{-b}{2a}) \cup (\frac{-b}{2a}, \infty) \]

### Conclusion
Thus, the solution set for the inequality \( ax^2 + bx + c > 0 \) with the given conditions is all real numbers except the double root.