If D be the discriminant of the quadratic equation `ax^2+bx+c=0` what will be the coordinates of its vertex?

Options

A. `-b//2a,-D//4a`
B. `-b//2a,D//4a`
C. `b//2a,-D//2a`
D. `b//2a,sqrtD//4a`

Answer

To find the coordinates of the vertex of the quadratic equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are the coefficients of the equation.

### Step 2: Use the formula for the x-coordinate of the vertex
The x-coordinate of the vertex of a parabola represented by the quadratic equation \( ax^2 + bx + c \) is given by the formula:
\[ x = -\frac{b}{2a} \]

### Step 3: Substitute the values of \( a \) and \( b \)
Substituting the values of \( a \) and \( b \) into the formula, we can find the x-coordinate of the vertex.

### Step 4: Calculate the y-coordinate of the vertex
To find the y-coordinate of the vertex, we substitute the x-coordinate back into the original quadratic equation:
\[ y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c \]
This simplifies to:
\[ y = a\left(\frac{b^2}{4a^2}\right) - \frac{b^2}{2a} + c \]
\[ y = \frac{b^2}{4a} - \frac{2b^2}{4a} + c \]
\[ y = -\frac{b^2}{4a} + c \]

### Step 5: Combine the coordinates
Thus, the coordinates of the vertex are:
\[ \left(-\frac{b}{2a}, -\frac{b^2}{4a} + c\right) \]

### Final Answer
The coordinates of the vertex of the quadratic equation \( ax^2 + bx + c = 0 \) are:
\[ \left(-\frac{b}{2a}, -\frac{b^2}{4a} + c\right) \]