Using Diagrams to Find Equivalent Quadratic Expressions

7.8.3 Using Diagrams to Find Equivalent Quadratic Expressions

7.8.3 • Using Diagrams to Find Equivalent Quadratic Expressions

Activity

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 $x^{2} + 4 x + 3$ are equivalent expressions.

An area model using a rectangle is divided into 4 smaller rectangles with both vertical and horizontal segments. The top left section of the rectangle is labeled x and the top right section of the rectangle is labeled 3. The top left side length of the rectangle is labeled x and the bottom left side length of the rectangle is labeled 1. There are four areas inside the rectangle that are not yet labeled.

Compare your answer:

$x \cdot x = x^{2}$

$1 \cdot x = x$

$x \cdot 3 = 3 x$

$1 \cdot 3 = 3$

Adding these together gives $x^{2} + 4 x + 3$ .

2. Draw diagrams to help you write an equivalent expression for each of the following:

a. $(x + 5)^{2}$

Compare your answer:

The diagram should show partial products that add up to $x^{2} + 10 x + 25$ .

b. $2 x (x + 4)$

Compare your answer:

The diagram should show partial products that add up to $2 x^{2} + 8 x$ .

c. $(2 x + 1) (x + 3)$

Compare your answer:

The diagram should show partial products that add up to $2 x^{2} + 7 x + 3$ .

d. $(x + m) (x + n)$

Compare your answer:

The diagram should show partial products that add up to $x^{2} + x n + m x + x n$ or $x^{2} + (m + n) x + m n$ .

3. Write an equivalent expression for each expression without drawing a diagram. Use the distributive property to help. Remember FOIL for multiplying binomials: Multiply the First terms, the Outer terms, the Inner terms, and the Last terms.

a. $(x + 2) (x + 6)$

Compare your answer:

$x^{2} + 8 x + 12$

b. $(x + 5) (2 x + 10)$

Compare your answer:

$2 x^{2} + 20 x + 50$

Are you ready for more?

Extending Your Thinking

Image 1

Individual algebra tiles are depicted. There is one quadratic tile that is labeled with an x and x along its top and left-hand sides. There is one linear tile labeled with a 1 and x along its top and left-hand side. In addition, are are also 4 other linear tiles that are not labeled. There are also 6 small unit tiles that are not labeled.

Image 2

Algebra tiles arranged into an algebraic model that is rectangular in shape. Along the bottom is one quadratic square that is labeled x and x along two of its sides. Stacked horizontally above the square are three linear tiles. Stacked vertically to the right of the square are two linear tiles. In the space formed by the vertically and horizontally arranged linear tiles, the six unit tiles are arranged to complete the rectangle.

Are Image 1 and Image 2 equivalent? Explain or show your reasoning.

Compare your answer:

Yes, because $x \cdot x + 5 (1 x) + 6 (1) = (x + 2) (x + 3)$ .

What does this tell you about an equivalent expression for $x^{2} + 5 x + 6$ ?

Compare your answer:

This shows that an equivalent expression is $(x + 2) (x + 3)$ .

Is there a different non-zero number of 1-by-1 squares that we could use instead that would allow us to arrange the combined figures into a single large rectangle?

Compare your answer:

We could use four 1-by-1 squares. This would give the expression $x^{2} + 5 x + 4$ . The rectangle would be $(x + 4) (x + 1)$ .

Video: Using Diagrams to Find Equivalent Quadratic Expressions

Watch the following video to learn more about using diagrams to find equivalent quadratic expressions.

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Self Check

Which expression is equivalent to

(x +2) (x + 4)

$x^2+2x+8$
$7x^2+8$
$x^2+6x+8$
$x^2+4x$

Additional Resources

Multiplying Binomials

Write an expression equivalent to $(x + 2) (x + 5)$ .

Step 1 - Distribute. Use FOIL. Multiply the First terms, Outer terms, Inner terms, and Last terms.

Step 2 - Write out the four terms.

$x^{2} + 5 x + 2 x + 10$

Step 3 - Combine like terms.

$x^{2} + 7 x + 10$

Try it

Multiplying Binomials

Write an expression equivalent to $(x + 7) (x + 3)$ .

Here is how to multiply binomial expressions:

Step 1 - Distribute.

A diagram showing the quantity of x plus 7 times the quantity of x plus 3. On the top, a red arrow is going from x in the first binomial to x in the second binomial. Another red arrow is going from x in the first binomial to 3 in the second binomial. On the bottom, in blue, an arrow is going from 7 in the first binomial to x in the second binomial and then another arrow is going from 7 in the first binomial to 3 in the second binomial.

Step 2 - Write out the four terms. $x^{2} + 3 x + 7 x + 10$

Step 3 - Combine like terms. $x^{2} + 10 x + 21$