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.)

Highlight the connections between the numbers in the solution and the parameters in the original equation.

• -5 is .
• is . 
• 12 is , or .
• 2 is , or . 

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). 

Some students may think that we multiply just by 4 and wonder where the comes from. Point out that we need the first term to be a perfect square. In the previous activity, the value was 1 (which was already a perfect square). We can’t be certain that is a perfect square, so we multiply by 4 and  to make the first term , which is , a perfect square.

Some may wonder where the comes from. Consider displaying a completed solution from the previous activity that parallels this work. Remind students of the placeholder used there. Invite them to compare other steps in the previous activity to those in this activity to help them explain the derivation.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their explanation for each step in the derivation of the quadratic formula. Tell students to focus on their reasoning for the second step and the step in which a constant is added to each side. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback:

• “_____ makes sense, but what do you mean when you say. . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

Invite students to share their explanations for each step. Highlight a few key moves:

• Multiplying the equation by makes the coefficient of a perfect square and the coefficient of the linear term an even number, both of which make completing the square much easier.

  Some students may wonder why is chosen to complete the square when solving . Why not , , or ? Any of these would work. The equation could also have been divided by to make the first term , which is a perfect square. (If time permits, consider asking students to complete the square using one of these alternatives.) While these alternatives result in a perfect square for the first term, they may not give an even number for the coefficient of the second term, or they may produce an equation with larger numbers, making it a bit trickier to manipulate.

• We can tell which constant term is needed to complete the square by dividing the coefficient of the linear term by the positive square root of the coefficient of , taking half of it, and squaring that number. In this case:
  • The coefficient of is .
  • The coefficient of is , so the square root is .
  • Dividing by gives . Half of is .
  • The constant term that completes the square is .

Emphasize that the quadratic formula essentially captures the steps for completing the square in one expression. Every time we solve a quadratic equation by completing the square, we are essentially using the quadratic formula, but in a less condensed way.