In this activity, students begin to complete the square. They start by transforming given perfect squares from standard form to factored form, and vice versa. Then, they are given an incomplete expression in standard form that contains only the squared term and linear term. Students need to decide what constant term makes the expression a perfect square, and then write the equivalent expression in factored form. To accomplish these tasks, students must rely on the structure they noticed in an earlier lesson about the relationship between the standard and factored forms of perfect squares (MP7).

If students have trouble determining the constant term in standard form, suggest that they draw a rectangular diagram and work backward to determine the factors along the two sides of the rectangle. Afterward, they can find the corresponding value of the constant term.

This diagram represents the expression (in standard form) and the expression (in factored form).

Display the incomplete table for all to see. Ask students to complete the missing values or expressions.

Discuss how students knew what numbers or expressions to write in the last four pairs of expressions. Make sure students understand that:

• If the constant term in the factored form is , the constant term in standard form is , a squared number.
• The coefficient of the linear term in standard form is twice of , or .
• This means that is half of the coefficient of the linear term. If the linear term is ,  is 10, and the constant term in standard form is , or 100.
• If the linear term in standard form is  (as in the last row of the table), then the constant term in standard form is , and the constant term in factored form is .

Explain to students that finding the constant term to add in order to create a perfect square is called “completing the square.”

Earlier, students identified that an expression in standard form can be written as a perfect square and learned to write it that way. Here, they learn to use that skill to solve quadratic equations.

Select students to share their solutions and to display their reasoning for all to see. After each student presents, ask if others found the same solution but completed the square in a different way. Make sure students see that the steps could vary, but the solutions should be the same if equality between the two sides of the equal sign is maintained throughout the solving process.

Ask students how they could check their solutions. One way is by substituting the solutions back into the equation and seeing if the equation is true at those values of the variable. For example, to see if -4 and -2 are the right solutions to the equation , we can evaluate and and see if each has a value of 0. Both and give 0, so the solutions are correct.