A.6.9.2 Finding Products of Differences
We have used area diagrams to expand expressions such as x(x + 5) and write equivalent expressions. When dealing with negative numbers, however, thinking in terms of finding area isn’t very helpful. We can still draw rectangular diagrams, but use them to organize the two factors and the results of applying the distributive property.
For example, this diagram shows that x(x + 5) = x2 + 5x.
These diagrams can be used even when subtraction is involved. To represent x(x - 5), we can rewrite it as
x(x + -5) then label the diagram as follows:
The diagram shows that x(x - 5) = x(x + -5) = x2 + -5x = x2 - 5x
Reminder - in an earlier lesson, we saw that a quadratic expression of the form (x + p)(x + q) is equivalent to: x2 + px + qx + pq or x2 + (p + q) = pq.
These diagrams can be used even when subtraction is involved. To represent x(x - 5), we can rewrite it as
x(x + -5) then label the diagram as follows:
The diagram shows that x(x - 5) = x(x + -5) = x2 + -5x = x2 - 5x
Reminder - in an earlier lesson, we saw that a quadratic expression of the form (x + p)(x + q) is equivalent to: x2 + px + qx + pq or x2 + (p + q) = pq.Show that (x - 1)(x - 1) and x2 - 2x + 1 are equivalent expressions by drawing a diagram and applying the distributive property. Explain your reasoning here and draw the diagram in the box below.
For the expression (x + 1)(x - 1) write an equivalent expression. Show your reasoning (either by drawing a diagram or applying the distributive property).
For the expression (x - 2)(x + 3) write an equivalent expression. Show your reasoning (either by drawing a diagram or applying the distributive property).
For the expression (x - 2)2 write an equivalent expression. Show your reasoning (either by drawing a diagram or applying the distributive property).