Lesson 8

Equivalent Quadratic Expressions

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

8.1: Diagrams of Products

Rectangle divided into 2 smaller rectangles. Top labeled 6. Left side labeled 3 and 4.
  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.

Rectangle, divided into 2 smaller rectangles. Top of left rectangle labeled x, top of right rectangle labeled 2. Side length labeled 4. Inside left rectangle, 4 x. Inside right rectangle, 8.
  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.
    Large rectangle, divided into 4 smaller rectangles. Left rectangle on top labeled x, Right rectangle on top labeled 3. Top rectangle on left side labeled x, bottom rectangle on left side labeled 1.
  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)


Square, side length = x. 5 rectangles, width = x, length = 1. 6 squares with side length = 1.
  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.

Rectangle divided into 4 smaller rectangles.

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).

Diagram showing distributive property.

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.