This Warm-up reminds students of the structure that governs the relationship between perfect squares written as squared factors and their equivalent expression in standard form. The key goal is for students to see that when we expand a squared factor of the form , the equivalent expression in standard form has this structure: . Seeing the coefficient of the squared term as and the coefficient of the linear term as 2 times a product of two numbers ( and ) enables students to complete the square more easily. This, in turn, helps them make sense of where the quadratic formula comes from, which they will explore in this lesson.

Students practice looking for and making use of structure as they think about the relationships between equivalent expressions (MP7).

Display the completed table for all to see. Invite students to share their solutions for the last question. Then, ask how these values can be used to complete the square.

• “When making a perfect square, how do you know what values to write in the middle column?” (The first squared term is  and equals , so that “something” must be  because . The linear term is  and equals , so that other thing must be 3 because . The last squared term is , so it is .)

Emphasize the general structure of the equivalent expressions, as shown in the last row. When  is expanded, the standard form always has the structure of . Also, make sure students understand how the numbers in the factored form are related to —that is, that they recognize that , , and .

Highlight that when we want to complete the square for and write an equivalent expression of the form , having a perfect square for makes it much easier to find , and having an even number for makes it easier to find (because an even number gives a whole number when divided by 2).

This is the first of two activities that help students derive the quadratic formula. Here students complete the square for a quadratic equation without evaluating the numerical expressions along the way. Doing so enables them to recognize the parts of the quadratic formula in numerical form (for example, instead of seeing , they see ) by comparing the structure of the solutions (MP7).

The linear coefficient of the given equation is an odd number. Students learn that there are strategies to transform the quadratic expression such that it has an even number for the linear coefficient and 1 for the leading coefficient, which makes it much easier to complete the square. These strategies involve multiplying the equation by a helpful factor and temporarily using simpler variables to stand in for complicated parts of an expression.

(The strategy of multiplying an equation by a number to make  a perfect square is illustrated in an optional activity in the last lesson on completing the square. Students who completed that activity may recall this approach. Using a simple variable to stand in for a messier expression is also explored in an optional activity in an earlier lesson—the last lesson on rewriting expressions in factored form.)

In the previous activity, students solved a quadratic equation by completing the square, but they did so without evaluating any of the numerical expressions. Students explained each step in the solving process and saw how it produced a solution that looks almost exactly like the quadratic formula, except that it contains numbers instead of and . That activity gave students concrete insights into the process of deriving the quadratic formula, preparing them to do the same in more abstract terms.

In this activity, students study a series of steps taken to solve by completing the square and make sense of how it leads to the quadratic formula. Along the way, they see that the solving process involves similar maneuvers as those seen earlier, for example, multiplying the equation by a factor to make the leading coefficient a perfect square, writing the linear coefficient as 2 times two numbers, and adding the square of one of the numbers to complete the square. They see that the result of solving the equation by completing the square is the quadratic formula.

By providing the explanations for each student, students are practicing constructing arguments (MP3).