- Indra B. Kshattry PhD

General quadratic into vertex form

Conversion of general quadratic Function $$ f(x) = ax^2 + bx + c $$ into its vertex form $$ f(x) = a(x-h)^2 + k $$ using the method of completing the square.

1. Start with the standard form:

   $$ f(x) = ax^2 + bx + c $$

2. Factor 𝑎 from the first two terms:

   $$ f(x) = a (x^2 + \frac{b}{a} x ) + c $$

3. Complete the square:

   To complete the square inside the parentheses, consider the quadratic term \(x^2 + \frac{b}{a} x\). The key step is to add and subtract \((\frac{Coefficient \quad of \quad x}{2})^2 \) $$ (\frac{Coefficient \quad of \quad x}{2})^2 = (\frac{\frac{b}{a} }{2})^2 = (\frac{b}{2a})^2 $$ Add and subtract this square term inside the parentheses: $$ f(x) = a (x^2 + \frac{b}{a} x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2 ) + c $$

4. Reorganize the terms inside the parentheses:

   Reorganize the terms inside the parentheses: $$ f(x) = a ((x + \frac{b}{2a})^2 - (\frac{b}{2a})^2 ) + c $$

5. Distribute 𝑎 across the terms:

   Expand 𝑎 into both terms inside the parentheses: $$ f(x) = a (x + \frac{b}{2a})^2 - a(\frac{b}{2a})^2 + c $$

6. Simplify the constant term:

   Simplify \( - a(\frac{b}{2a})^2 \) \[ -a\left(\frac{b}{2a}\right)^2 = -a \cdot \frac{b^2}{4a^2} = -\frac{b^2}{4a} \] Thus, the equation becomes: $$ f(x) = a (x + \frac{b}{2a})^2 + c -\frac{b^2}{4a} $$

7. Combine constants:

   Combine 𝑐 and - \(\frac{b^2}{4a}\) into a single constant term: $$ f(x) = a (x + \frac{b}{2a})^2 + (c -\frac{b^2}{4a}) $$

8. Rewrite in vertex form:

   The vertex form is: $$ f(x) = a(x - h)^2 + k $$ Here,
   • \(h = - \frac{b}{2a}\), which represents the x-coordinate of the vertex.
   • \(k = c -\frac{b^2}{4a}\), which represents the y-coordinate of the vertex.
   Final vertex form: $$ f(x) = a (x + \frac{b}{2a})^2 + \frac{4ac - b^2}{4a} $$

This shows the detailed conversion of the standard form \( f(x) = ax^2 + bx + c\) into the vertex form \(f(x) = a(x - h)^2 + k\)