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

Options

A. Least value `-(1)/(2)`
B. Least value `(9)/(4)`
C. Greastest value `(1)(4)`
D. Greatest value `(9)/(4)`

Answer

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

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

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation of the form \( ax^2 + bx + c = 0 \):
- The sum of the roots (\( \alpha + \beta \)) is given by \( -\frac{b}{a} \).
- The product of the roots (\( \alpha \beta \)) is given by \( \frac{c}{a} \).

In our equation:
- \( a = 1 \)
- \( b = \alpha \)
- \( c = -\beta \)

Thus, we have:
1. Sum of roots:
\[ \alpha + \beta = -\alpha \]
which simplifies to:
\[ \beta = -2\alpha \]

2. Product of roots:
\[ \alpha \beta = -\beta \]
which simplifies to:
\[ \alpha \beta + \beta = 0 \]
or:
\[ \beta(\alpha + 1) = 0 \]

Since \(\beta \neq 0\), we have:
\[ \alpha + 1 = 0 \]
Thus,
\[ \alpha = -1 \]

### Step 3: Find the value of \(\beta\)
Substituting \(\alpha = -1\) into the equation \(\beta = -2\alpha\):
\[ \beta = -2(-1) = 2 \]

### Step 4: Substitute \(\alpha\) and \(\beta\) into the new quadratic expression
Now we substitute \(\alpha\) and \(\beta\) into the expression:
\[ -x^2 + \alpha x + \beta \]
This becomes:
\[ -x^2 - x + 2 \]

### Step 5: Determine the maximum value of the quadratic expression
The expression \(-x^2 - x + 2\) is a downward-opening parabola (since the coefficient of \(x^2\) is negative). The maximum value can be found using the vertex formula:
The x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]
where \(a = -1\) and \(b = -1\):
\[ x = -\frac{-1}{2 \cdot -1} = \frac{1}{2} \]

### Step 6: Substitute \(x = \frac{1}{2}\) back into the expression
Now we substitute \(x = \frac{1}{2}\) into the expression:
\[ -\left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 2 \]
Calculating this:
\[ -\frac{1}{4} - \frac{1}{2} + 2 = -\frac{1}{4} - \frac{2}{4} + \frac{8}{4} = \frac{5}{4} \]

### Step 7: Conclusion
Thus, the maximum value of the expression \(-x^2 + \alpha x + \beta\) is:
\[ \frac{9}{4} \]

### Final Answer
The greatest value of the quadratic expression is \( \frac{9}{4} \).

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