Sometimes a quadratic relationship can be expressed without writing a squared term that appears as a variable raised to the second power (like  or ). Let’s take this pattern of squares, for example.

Three steps of a growing pattern. Step 1: Total of two squares, stacked one atop the other. Step 2: Total of six squares, two squares on row 1, row 2 and row 3. Step 3: Total of twelve squares, three squares on row 1, row 2, row 3 and row 4.

From the first 3 steps, we can see that both the length and the width of the rectangle increase by 1 at each step. Step 1 is a 1-by-2 rectangle, Step 2 is a 2-by-3 rectangle, and Step 3 is a 3-by-4 rectangle. This suggests that Step is a rectangle with side lengths of  and , so the number of squares at Step is .

This expression may not look like quadratic expressions with a squared term, which we saw in earlier lessons, but if we apply the distributive property, we can see that is equivalent to .

We can also visually show that these expressions are the equivalent by breaking each rectangle into an -by- square (the in the expression) and an -by- rectangle (the in the expression).

Three steps of a growing pattern. Step 1: Total of two squares, stacked one above the other with a space between the rows. Step 2: Total of six squares, two squares on row 1, row 2 and row 3. Row 3 is above the others with a space between rows 2 and 3. Step 3: Total of twelve squares, three squares on row 1, row 2, row 3 and row 4. Row 4 is above the others with a space between rows 3 and 4.

The relationship between the step number and the number of squares can be described by a quadratic function, , whose input is and whose output is the number of squares at Step . We can define with or with  .