Each term in the expression represents part of a side in the generic rectangle.
B
b
Note that a^2=a* a.
A
a
12x^2+17x-5
B
b
4x^2-28x+49
Practice makes perfect
a
Each term in our expression represents a part of a side in the generic rectangle. With this information, we can draw the following diagram.
Let's calculate the products.
By adding the areas of the smaller rectangles we get the area of the generic rectangle. This is the same as the product of its sides.
(4x-1)(3x+5)=12x^2+20x-3x-5
⇓
(4x-1)(3x+5)=12x^2+17x-5
To verify Casey's pattern we will multiply the expressions across the diagonals of the rectangle.
If the product of the numbers across one diagonal equals the product of the numbers across the other diagonal, the pattern is correct.
Squaring a number or expression is the same as multiplying it by itself. We can rewrite our expression into multiplication.
(2x-7)^2=(2x-7)(2x-7)
Now we can create our generic rectangle.
Let's calculate the products.
By adding the areas of the smaller rectangles we get the area of the generic rectangle. This is the same as the product of its sides.
(2x-7)^2=4x^2+(- 14x)+(- 14x)+49
⇓
(2x-7)^2=4x^2-28x+49
To verify Casey's pattern we will multiply the expressions across the diagonals of the rectangle.
If the product of the numbers across one diagonal equals the product of the numbers across the other diagonal, the pattern is correct.
