IM Alg1.6.8 Lesson: Equivalent Quadratic Expressions

[img]data:image/png;base64,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[/img][br][br]Explain why the diagram shows that [math]6\left(3+4\right)=6\cdot3+6\cdot4[/math].[br]

Draw a diagram to show that 5(x+2)=5x+10.

[size=150]Applying the distributive property to multiply out the factors of, or expand, [math]4\left(x+2\right)[/math] gives us [math]4x+8[/math], so we know the two expressions are equivalent. We can use a rectangle with side lengths [math]\left(x+2\right)[/math] and 4 to illustrate the multiplication.[/size][br][br][img]data:image/png;base64,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[/img][br]

Draw a diagram to show that n(2n+5) and 2n²+5n are equivalent expressions.

For each expression, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.[br][br][list][*][math]6\left(\frac{1}{3}n+2\right)[/math][/*][/list]

[list][*][math]p\left(4p+9\right)[/math][/*][/list]

[list][*][math]5r\left(r+\frac{3}{5}\right)[/math][/*][/list]

[list][*][math]\left(0.5w+7\right)w[/math][/*][/list]

Here is a diagram of a rectangle with side lengths x+1 and x+3.

[size=150]Use this diagram to show that [math]\left(x+1\right)\left(x+3\right)[/math] and [math]x^2+4x+3[/math] are equivalent expressions.[/size]

Draw diagrams to help you write an equivalent expression for each of the following: (x+5)²

Draw diagrams to help you write an equivalent expression for each of the following: 2x(x+4)

Draw diagrams to help you write an equivalent expression for each of the following: (2x+1)(x+3)

Draw diagrams to help you write an equivalent expression for each of the following: (x+m)(x+n)

[size=150]Write an equivalent expression for each expression without drawing a diagram:[/size][br][br][math]\left(x+2\right)\left(x+6\right)[/math]

[math]\left(x+5\right)\left(2x+10\right)[/math]

Is it possible to arrange an [math]x[/math] by [math]x[/math] square, five [math]x[/math] by 1 rectangles and six 1 by 1 squares into a single large rectangle? Explain or show your reasoning.[br]

What does this tell you about an equivalent expression for [math]x^2+5x+6[/math]?

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?[br]