a. A quadratic expression in the form $a^{2} x^{2}-b^{2}$ is called a difference of squares. Use a generic rectangle to prove that $a^{2} x^{2}-b^{2}=(a x-b)(a x+b)$ . Be ready to share your work with the class. $\newline$ b. A quadratic expression in the form $a^{2} x^{2}+2 a b x+b^{2}$ is called a perfect square trinomial. Use a generic rectangle to prove that $a^{2} x^{2}+2 a b x+b^{2}=(a x+b)^{2}$ . Be ready to share your work with the class.

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Algebra 2

Reflections of functions

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Q. a. A quadratic expression in the form

a^{2} x^{2}-b^{2}

is called a difference of squares. Use a generic rectangle to prove that

a^{2} x^{2}-b^{2}=(a x-b)(a x+b)

. Be ready to share your work with the class.

\newline

b. A quadratic expression in the form

a^{2} x^{2}+2 a b x+b^{2}

is called a perfect square trinomial. Use a generic rectangle to prove that

a^{2} x^{2}+2 a b x+b^{2}=(a x+b)^{2}

. Be ready to share your work with the class.

Draw Rectangle Sections: Draw a generic rectangle with four sections for the difference of squares.
Label Length and Width: Label the length of the rectangle as $a$ for the top two sections, and the width as $a$ for the top two sections, and $-b$ and $b$ for the bottom two sections.
Fill in Areas: Fill in the areas of each section: top left is $a^{2}x^{2}$ , top right is $abx$ , bottom left is $-abx$ , and bottom right is $-b^{2}$ .
Combine Areas: Combine the areas to form the expression: $a^{2}x^{2} + abx - abx - b^{2}$ .
Cancel Out Terms: Notice that $abx$ and $-abx$ cancel each other out, leaving $a^{2}x^{2} - b^{2}$ .
Calculate Area: The length of the rectangle is $(ax + b)$ and the width is $(ax - b)$ , so the area is $(ax - b)(ax + b)$ .
Draw Another Rectangle: Now, draw another generic rectangle for the perfect square trinomial.
Label Length and Width: Label the length of the rectangle as $ax$ and the width as $ax$ for the top two sections, and $b$ and $b$ for the bottom two sections.
Fill in Areas: Fill in the areas of each section: top left is $a^{2}x^{2}$ , top right is $abx$ , bottom left is $abx$ , and bottom right is $b^{$2$}.
Combine Areas: Combine the areas to form the expression: $a^{2}x^{2} + abx + abx + b^{2}$.
Calculate Area: Notice that $abx$ and $abx$ add up to $2abx$, leaving $a^{2}x^{2} + 2abx + b^{2}$.
Calculate Area: Notice that $abx$ and $abx$ add up to $2abx$, leaving $a^{2}x^{2} + 2abx + b^{2}$.The length of the rectangle is $(ax + b)$ and the width is also $(ax + b)$, so the area is $(ax + b)(ax + b)$ or $(ax + b)^{2}$.