# Lesson 8

• Let’s use diagrams to help us rewrite quadratic expressions.

### 8.1: Diagrams of Products

1. Explain why the diagram shows that $$6(3+4) = 6 \boldcdot 3 + 6 \boldcdot 4$$.
2. Draw a diagram to show that $$5(x+2) = 5x + 10$$.

### 8.2: Drawing Diagrams to Represent More Products

Applying the distributive property to multiply out the factors of, or expand, $$4(x+2)$$ gives us $$4x + 8$$, so we know the two expressions are equivalent. We can use a rectangle with side lengths $$(x+2)$$ and 4 to illustrate the multiplication.

1. Draw a diagram to show that $$n(2n+5)$$ and $$2n^2 + 5n$$ are equivalent expressions.
2. For each expression, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.

a. $$6\left(\frac13 n + 2\right)$$

b. $$p(4p + 9)$$

c. $$5r\left(r + \frac35\right)$$

d. $$(0.5w + 7)w$$

### 8.3: Using Diagrams to Find Equivalent Quadratic Expressions

1. Here is a diagram of a rectangle with side lengths $$x+1$$ and $$x+3$$. Use this diagram to show that $$(x+1)(x+3)$$ and $$x^2 + 4x+3$$ are equivalent expressions.
2. Draw diagrams to help you write an equivalent expression for each of the following:
1. $$(x+5)^2$$
2. $$2x(x+4)$$
3. $$(2x+1)(x+3)$$
4. $$(x+m)(x+n)$$
3. Write an equivalent expression for each expression without drawing a diagram:
1. $$(x +2)(x + 6)$$
2. $$(x +5)(2x + 10)$$

1. Is it possible to arrange an $$x$$ by $$x$$ square, five $$x$$ by 1 rectangles and six 1 by 1 squares into a single large rectangle?  Explain or show your reasoning.
2. What does this tell you about an equivalent expression for $$x^2 + 5x + 6$$?
3. Is there a different non-zero number of 1 by 1 squares that we could have used instead that would allow us to arrange the combined figures into a single large rectangle?

### Summary

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 $$18x - x^2$$. The former is a product of $$x$$ and $$18-x$$, and the latter is a difference of $$18x$$ and $$x^2$$, but both expressions represent the same function.

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, and each side length decomposed into a variable expression and a number.

Multiplying $$(x+2)$$ and $$(x+3)$$ gives the area of the rectangle. Adding the areas of the four sub-rectangles also gives the area of the rectangle. This means that $$(x+2)(x+3)$$ is equivalent to $$x^2 + 2x + 3x + 6$$, or to $$x^2 + 5x + 6$$.

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 $$x^2 + (p+q)x + pq$$.