Writing Equivalent Expressions

7.8.4 Writing Equivalent Expressions

7.8.4 • Writing Equivalent Expressions

Cool Down

Draw a diagram to show that $(3 x + 1) (x + 2)$ is equivalent to $3 x^{2} + 7 x + 2$ .

Compare your answer:

Diagram should show partial products of $3 x (x)$ , $6 x$ , $x$ , and 2, which add up to $3 x^{2} + 7 x + 2$ .

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 2. The top left side length of the rectangle is labeled 3 times x and the bottom left side length of the rectangle is labeled 1. There are four areas inside the rectangle: The top left area that is bordered by x and 3x is labled 3 times x squared. The top right area under the 2 is labeled 6 times x. In the bottom row, the left region is labeled x and the right region is labeled 2.

Is $(x + 4)^{2}$ equivalent to $2 x^{2} + 8 x + 8$ ? Explain or show your reasoning.

Compare your answer:

No. $(x + 4) (x + 4) = x^{2} + 4 x + 4 x + 16 = x^{2} + 8 x + 16$ .

Why Should I Care?

Illustration of a vibrant pumpkin patch with scattered pumpkins, a wheelbarrow, and a "Pumpkin" sign. In the background, barns and a windmill stand under cloudy skies.

Anna's class entered a contest where they had to build a catapault and use it to launch a pumpkin at a distant target. But when Anna's class shot their pumpkin for the first time, it went wildly off course.

Anna thought about their problem and came up with a solution. If Anna's class used a quadratic equation, they could mathematically find the best place and angle for their pumpkin catapault. It worked! Their accuracy went from 2% to 99%.

You can use quadratic equations for any object that travels in a curved path, like footballs, frisbees, and golf balls.