Lesson 7 Square Deal Solidify Understanding
Find a process for completing the square that works on all quadratic functions.
Adapt diagrams to become more efficient in completing the square.
How can we complete the square when there is more than one square given (or
Open Up the Math: Launch, Explore, Discuss
Remember Optima’s quilt shop? She bases her designs on quilt squares that can vary in size, so she calls the length of the side for the basic square
Sometimes a customer orders more than one quilt block of a given size. For instance, when a customer orders
One of the customer service representatives finds an envelope that contains the blocks pictured below. Write the order that shows two equivalent equations for the area of the blocks.
What equations for the area could customer service write if they received an order for
What if customer service receives an order for
Clementine is at it again! When is that dog going to learn not to chew up the orders? (She also chews Optima’s shoes, but that’s a story for another day.) Here are some of the orders that have been chewed up so they are missing the last term. Help Optima by completing each of the following expressions for the area so that they describe a perfect square. Then, write the two equivalent equations for the area of the square.
Sometimes the quilt shop gets an order that turns out to be more or less than a perfect square. Customer service always tries to fill orders with perfect squares, or at least, they start there and then adjust as needed. They always write their equations in a way that relates the area to the closest perfect square.
Now here’s a real mess! Customer service received an order for an area
Optima really needs to get organized. Here’s another scrambled diagram. Write two equivalent equations for the area of this diagram.
Optima realized that not everyone needs perfect squares and not all orders are coming in as expressions that are perfect squares. Determine whether each expression below is a perfect square and why the expression is or is not a perfect square. If it is not a perfect square, find the perfect square that seems “closest” to the given expression, and show how the perfect square can be adjusted to be equivalent to the given expression.
Now let’s generalize. Given an expression in the form
Ready for More?
Use the completing the square process on the following expression:
Completing the square for
Completing the square for
In this lesson, we solidified a process for completing the square with expressions in the form
Find the indicated values for