A quadratic function can often be represented by many equivalent expressions. For example, a quadratic function, $f$, might be defined by $f(x) = x^2 + 3x + 2$. The quadratic expression $x^2 + 3x + 2$ is called the standard form, the sum of a multiple of $x^2$ and a linear expression ($3x+2$ in this case).

In general, standard form is written as $\displaystyle ax^2 + bx + c$ 

We refer to $a$ as the coefficient of the squared term $x^2$, $b$ as the coefficient of the linear term $x$, and $c$ as the constant term.

Function $f$ can also be defined by the equivalent expression $(x+2)(x+1)$. 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 $(x+3)(x+2)$. We can do the same to expand an expression with a sum and a difference, such as $(x+5)(x-2)$, or to expand an expression with two differences, for example, $(x-4)(x-1)$.

To represent $(x-4)(x-1)$ with a diagram, we can think of subtraction as adding the opposite:

	$x$	$\text-4$
$x$	$x^2$	$\text-4x$
$\text-1$	$\text-x$	$4$

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.