How can you use the values of a, b, and c to write a quadratic function in vertex form?

y = a (x + b / (2a)) ^2 + (4ac - b^2) / (4a)

Step-by-step explanation:

We presume you're starting with ...

y = ax^2 + bx + c

As you would with numbers, factor the first two terms:

y = a (x^2 + b/a·x) + c

Add half the square of the x term inside and its opposite outside parentheses:

y = a (x^2 + (b/a) x + (b / (2a)) ^2) + c - a (b / (2a)) ^2

y = a (x + b / (2a)) ^2 + c - b^2 / (4a)

You can combine the last two terms to a more familiar fraction:

y = a (x + b / (2a)) ^2 + (4ac - b^2) / (4a)