Q. directrix of parabola \(y = ax^2 + bx + c\) formula

Find the directrix of the parabola y = ax^2 + bx + c

We assume a ≠ 0. Below is a detailed, step-by-step derivation using completing the square and the standard parabola form.

1. Write the quadratic in vertex (completed square) form.

   Start with

   \[ y = ax^2 + bx + c. \]

   Factor \(a\) from the x-terms:

   \[ y = a\left(x^2 + \frac{b}{a}x\right) + c. \]

   Complete the square inside the parentheses. Add and subtract \(\left(\dfrac{b}{2a}\right)^2\):

   \[ y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 – \left(\frac{b}{2a}\right)^2\right) + c. \]

   Group the perfect square and simplify:

   \[ y = a\left(\left(x + \frac{b}{2a}\right)^2 – \left(\frac{b}{2a}\right)^2\right) + c. \]

   Distribute \(a\) and simplify the constant term:

   \[ y = a\left(x + \frac{b}{2a}\right)^2 – a\left(\frac{b}{2a}\right)^2 + c. \]

   Compute the squared term coefficient:

   \[ a\left(\frac{b}{2a}\right)^2 = a\cdot\frac{b^2}{4a^2} = \frac{b^2}{4a}. \]

   Thus the vertex form is

   \[ y = a\left(x + \frac{b}{2a}\right)^2 + \left(c – \frac{b^2}{4a}\right). \]

2. Identify the vertex (h, k).

   Compare with the vertex form \(y = a(x – h)^2 + k\). From the expression above we read

   \[ h = -\frac{b}{2a}, \qquad k = c – \frac{b^2}{4a}. \]

3. Convert to the standard parabola form to find p.

   The standard vertical-parabola form is

   \[ (x – h)^2 = 4p(y – k), \]

   so solve for \((x – h)^2\) in terms of \((y – k)\) using the vertex form:

   \[ y = a(x – h)^2 + k \quad\Longleftrightarrow\quad (x – h)^2 = \frac{1}{a}(y – k). \]

   Compare with \((x – h)^2 = 4p(y – k)\). Therefore

   \[ 4p = \frac{1}{a} \quad\text{so}\quad p = \frac{1}{4a}. \]

4. Write the directrix in terms of h, k, p.

   For a vertical parabola with vertex \((h,k)\) and parameter \(p\), the directrix is

   \[ y = k – p. \]

5. Substitute k and p expressed in a, b, c.

   Substitute \(k = c – \dfrac{b^2}{4a}\) and \(p = \dfrac{1}{4a}\) into \(y = k – p\):

   \[ y = \left(c – \frac{b^2}{4a}\right) – \frac{1}{4a}. \]

   Combine the fractional terms over the common denominator \(4a\):

   \[ y = c – \frac{b^2 + 1}{4a}. \]

Final formula for the directrix:

\[ \text{Directrix: } y = c – \dfrac{b^{2} + 1}{4a}, \quad \text{for } a \ne 0. \]

Optionally: verify with \(y = x^2\) (\(a=1, b=0, c=0\)) to obtain \(y = -\tfrac{1}{4}\), which matches the known result.