If `a and b(!=0)` are the roots of the equation `x^2+ax+b=0` then the least value of `x^2+ax+b` is

Options

A. `9/4`
B. `-(9)/4`
C. `-(1)/(4)`
D. `(1)/4`

Answer

To find the least value of the expression \( x^2 + ax + b \) given that \( a \) and \( b \) are the roots of the equation \( x^2 + ax + b = 0 \), we can follow these steps:

### Step 1: Understand the relationship between the coefficients and the roots
Given that \( a \) and \( b \) are the roots of the quadratic equation, we can use Vieta's formulas:
- The sum of the roots \( a + b = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -a \)
- The product of the roots \( ab = \frac{\text{constant term}}{\text{coefficient of } x^2} = b \)

### Step 2: Set up the equations from Vieta's formulas
From the sum of the roots:
\[ a + b = -a \implies b = -2a \]
From the product of the roots:
\[ ab = b \]
Since \( b \neq 0 \), we can divide both sides by \( b \):
\[ a = 1 \]

### Step 3: Substitute \( a \) back to find \( b \)
Now substituting \( a = 1 \) into the equation \( b = -2a \):
\[ b = -2(1) = -2 \]

### Step 4: Rewrite the expression with the values of \( a \) and \( b \)
Now we substitute \( a \) and \( b \) into the expression \( x^2 + ax + b \):
\[ x^2 + 1x - 2 = x^2 + x - 2 \]

### Step 5: Find the minimum value of the quadratic expression
To find the minimum value of the quadratic function \( f(x) = x^2 + x - 2 \), we can use the vertex formula. The x-coordinate of the vertex of a parabola given by \( ax^2 + bx + c \) is:
\[ x = -\frac{b}{2a} \]
Here, \( a = 1 \) and \( b = 1 \):
\[ x = -\frac{1}{2 \cdot 1} = -\frac{1}{2} \]

### Step 6: Calculate the minimum value by substituting back into the function
Now substitute \( x = -\frac{1}{2} \) back into the function \( f(x) \):
\[ f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) - 2 \]
Calculating this:
\[ = \frac{1}{4} - \frac{1}{2} - 2 \]
Convert \( -\frac{1}{2} \) and \( -2 \) to have a common denominator:
\[ = \frac{1}{4} - \frac{2}{4} - \frac{8}{4} = \frac{1 - 2 - 8}{4} = \frac{-9}{4} \]

### Conclusion
The least value of \( x^2 + ax + b \) is:
\[ \boxed{-\frac{9}{4}} \]