A quadratic function can often be defined by many different but equivalent expressions. For example, we saw earlier that the predicted revenue, in thousands of dollars, from selling a downloadable movie at $x$ dollars can be expressed with $x(18-x)$, which can also be written as $18x - x^2$.

Sometimes a quadratic expression is a product of two factors that are each a linear expression, for example $(x+2)(x+3)$. We can write an equivalent expression by thinking about each factor, the $(x+2)$ and $(x+3)$, as the side lengths of a rectangle, with each side length being decomposed into a variable expression and a number.

Rectangle, divided into 4 smaller rectangles. Top side of rectangle labeled x and 2. Left side of rectangle labeled x and 3. Small rectangles labeled as follows: Top left, x squared. Top right, 2 x. Bottom right, 6. Bottom left, 3 x.

Notice that the diagram illustrates the distributive property being applied. Each term of one factor (say, the $x$ and the 2 in $x+2$) is multiplied by every term in the other factor (the $x$ and the 3 in $x+3$).

In general, when a quadratic expression is written in the form of $(x+p)(x+q)$, we can apply the distributive property to rewrite it as $x^2 + px + qx + pq$, or as $x^2 + (p+q)x + pq$.