The most common way to write a quadratic function is to use general form:

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

When analyzing the graph of a quadratic function, or the correspondence between the graph and solutions to quadratic equations, two other forms are more suitable: vertex form and factor form.

In order to rewrite a quadratic function from general form to vertex form, you must know how to complete the square.

## Completing the square

Completing the square is a method used for solving quadratic equations and for rewriting quadratic functions to vertex form.

It is easy to solve a quadratic equation of the form $$x^2=c$$, you just take the square root of both sides. You can also easily solve equations of the form $$(ax+b)^2=c$$, first take the square root of both sides, then solve for $$x$$. When completing the square, you rewrite a quadratic equation to the form $$(ax+b)^2=c$$.

Let's say we want to solve an equation of the form

$x^2+bx = a$

where both $$a$$ and $$b$$ are positive numbers. In this case, we can use areas to explain how to complete the square.

From the worksheet we get that

\begin{align} x^2+bx &= a \\ x^2+bx + \left(\frac{b}{2}\right)^2 &= a + \left(\frac{b}{2}\right)^2 \\ \left(x+\frac{b}{2}\right)^2 &= a + \left(\frac{b}{2}\right)^2 \end{align}

which is an equation we can solve.

It is possible to make a similar geometric construction for negative values of $$b$$, but as long as we use a geometrical interpretation of the problem, $$a$$ must not be negative and the geometrical solution will not yield negative $$x$$-values. In order to get all roots in the general case, it is better to use an algebraic approach.

The first person to find a method for solving quadratic functions was Brahmagupta, who also made one of the greatest contributions to mankind ever ‐ he showed how to do arithmetic calculations using the number zero.

## Vertex form

A quadratic function can always be written as

$f(x) = a(x-h)^2+k$

which is called vertex form.

### Translations of graphs

The graph of a quadratic function $$g(x)=(x-h)^2+k$$, can be seen as the translation of the graph of the function $$f(x)=x^2$$ along the vector

$\binom{h}{k}.$

### From general form to vertex form

To rewrite a quadratic function from general to vertex form, you can complete the square. In the general we get this:

\begin{align} f(x) &= ax^2+bx+c \\ &= a\left( x^2+\frac{b}{a}x+\frac{c}{a}\right) \\ &= a\left( x^2+2\frac{b}{2a}x+ \left(\frac{b}{2a} \right)^2-\left(\frac{b}{2a} \right)^2+\frac{c}{a}\right) \\ &= a\left( \left(x + \frac{b}{2a}\right)^2 -\left(\frac{b}{2a} \right)^2+\frac{c}{a}\right) \\ &= a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a} \end{align}

## Factor form

A quadratic function can always be written as

$f(x) = a(x-\alpha)(x-\beta$

which is called vertex form. The coefficients $$\alpha$$ and $$\beta$$ are roots to the quadratic equation $$f(x) = 0$$.