The least value of `ax^2+bx+c,(a>0)` is

Options

A. `b^2-4ac`
B. `(-b)/(2a)`
C. `(-b)/a`
D. `(4ac-b^2)/(4a)`.

Answer

To find the least value of the quadratic expression \( ax^2 + bx + c \) where \( a > 0 \), we can follow these steps:

### Step-by-step Solution:

1. **Identify the Quadratic Function**:
The given expression is \( f(x) = ax^2 + bx + c \).

2. **Determine the Vertex of the Parabola**:
Since \( a > 0 \), the parabola opens upwards. The vertex of the parabola, which gives the minimum value, can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]

3. **Substitute the x-coordinate into the Function**:
To find the minimum value of the function, substitute \( x = -\frac{b}{2a} \) back into the function \( f(x) \):
\[ f\left(-\frac{b}{2a}\right) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c \]

4. **Simplify the Expression**:
Now simplify the expression:
\[ f\left(-\frac{b}{2a}\right) = a\left(\frac{b^2}{4a^2}\right) - \frac{b^2}{2a} + c \]
\[ = \frac{b^2}{4a} - \frac{2b^2}{4a} + c \]
\[ = \frac{b^2 - 2b^2}{4a} + c \]
\[ = -\frac{b^2}{4a} + c \]

5. **Express in Terms of Discriminant**:
The discriminant \( D \) of the quadratic is given by \( D = b^2 - 4ac \). Therefore, we can express the minimum value as:
\[ \text{Minimum value} = c - \frac{D}{4a} \]
where \( D = b^2 - 4ac \).

6. **Final Result**:
Thus, the least value of \( ax^2 + bx + c \) when \( a > 0 \) is:
\[ \text{Least Value} = -\frac{D}{4a} + c = \frac{4ac - b^2}{4a} \]

### Summary:
The least value of the quadratic expression \( ax^2 + bx + c \) (where \( a > 0 \)) is given by:
\[ \text{Least Value} = \frac{4ac - b^2}{4a} \]