A quadratic function can often be represented by many equivalent expressions. For example, a quadratic function, , might be defined by . The quadratic expression is called the standard form, the sum of a multiple of and a linear expression ( in this case).

In general, standard form is written as  

We refer to as the coefficient of the squared term , as the coefficient of the linear term , and as the constant term.

Function can also be defined by the equivalent expression . When the quadratic expression is a product of two factors where each one is a linear expression, this is called the factored form.

An expression in factored form can be rewritten in standard form by expanding it, which means multiplying out the factors. In a previous lesson we saw how to use a diagram and to apply the distributive property to multiply two linear expressions, such as . We can do the same to expand an expression with a sum and a difference, such as , or to expand an expression with two differences, for example, .

To represent with a diagram, we can think of subtraction as adding the opposite:

Diagram showing distributive property. Row 1: x minus four times x minus 1. Row 2: equals x plus negative 4 times x plus negative 1. Two arrows drawn from both first x and from negative 4, for each, one arrow to the second x, one arrow to negative 1. Row 3: equals x times the quantity x plus negative one, plus negative 4 times the quantity x plus negative 1. 2 arrows drawn from first x to second x and negative 1. 2 arrows drawn from negative 4 to third x and negative 1. Row 4: equals x squared plus negative 1 x plus negative 4 x plus negative 4 times negative 1. Row 5: equals x squared plus negative 5 x plus 4. Row 6: equals x squared minus 5 x plus 4.