If `alpha and beta ( != 0)` are the roots of the quadratic equation `x^(2) + ax - beta = 0," then the quadratic expression " -x^(2) + alpha x + beta " where " x in R `has

Options

A. Least value `-1/4`
B. Least value `-9/4`
C. Greatest value `1/4`
D. Greatest value `9/4`

Explanation

To solve the problem step by step, we will analyze the given quadratic equation and the expression we need to evaluate.

### Step 1: Identify the roots of the quadratic equation
The given quadratic equation is:
\[ x^2 + ax - \beta = 0 \]
with roots \( \alpha \) and \( \beta \).

### Step 2: Use Vieta's formulas
From Vieta's formulas, we know:
1. The sum of the roots \( \alpha + \beta = -a \)
2. The product of the roots \( \alpha \beta = -\beta \)

### Step 3: Rearranging the product of roots
From the product of roots:
\[ \alpha \beta = -\beta \]
Since \( \beta \neq 0 \), we can divide both sides by \( \beta \):
\[ \alpha = -1 \]

### Step 4: Substitute \( \alpha \) back to find \( \beta \)
Now substituting \( \alpha = -1 \) into the sum of roots:
\[ -1 + \beta = -a \]
This gives:
\[ \beta = -a + 1 \]

### Step 5: Write the new quadratic expression
The new quadratic expression we need to analyze is:
\[ -x^2 + \alpha x + \beta \]
Substituting \( \alpha \) and \( \beta \):
\[ -x^2 - x + (-a + 1) \]
This simplifies to:
\[ -x^2 - x + 1 - a \]

### Step 6: Identify the type of quadratic expression
This is a downward-opening parabola (since the coefficient of \( x^2 \) is negative). Therefore, it will have a maximum value.

### Step 7: Find the vertex of the parabola
The x-coordinate of the vertex (which gives the maximum value) can be found using:
\[ x = -\frac{b}{2a} \]
Here, \( a = -1 \) and \( b = -1 \):
\[ x = -\frac{-1}{2 \cdot -1} = \frac{1}{2} \]

### Step 8: Substitute \( x \) back into the expression to find the maximum value
Now substitute \( x = \frac{1}{2} \) back into the expression:
\[ y = -\left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + (1 - a) \]
Calculating this:
\[ y = -\frac{1}{4} - \frac{1}{2} + 1 - a \]
\[ y = -\frac{1}{4} - \frac{2}{4} + \frac{4}{4} - a \]
\[ y = \frac{1}{4} - a \]

### Step 9: Conclusion
Thus, the maximum value of the expression \( -x^2 + \alpha x + \beta \) is:
\[ \frac{1}{4} - a \]