Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

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.

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

1.

Image 1

Image 2

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

Compare your answer:

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

2.

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

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.

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

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

7.8.3 Using Diagrams to Find Equivalent Quadratic Expressions

Activity

Are you ready for more?

Extending Your Thinking

Video: Using Diagrams to Find Equivalent Quadratic Expressions

Additional Resources

Multiplying Binomials

Multiplying Binomials