The purpose of this activity is for students to analyze two valid methods of solving a quadratic equation (completing the square and the quadratic formula) that result in equivalent answers that look different. This gives students the opportunity to review the two methods. Students make viable arguments and critique the reasoning of others as they inspect the logic of each method and decide whether the different-looking answers are equivalent (MP3).

If students do not yet understand why implies when analyzing Elena’s work, consider asking:

• “Tell me more about how Elena solved her equation.”

• “What is the same and what is different about and ?”

The purpose of the discussion is to review methods for solving quadratic equations. Invite students to explain Elena’s and Han’s solutions. If these explanations do not come up, ask students:

• “How did Elena know that the left side of the equation could be written as from the beginning?” (The coefficient of is 1 and the coefficient of the linear term is -6. Half of -6 is -3, so completing the square correctly should result in the expression .)
• “Why did Elena add 2 to each side?” (To complete the square she needs to add or subtract a number to adjust the constant to be or 9. Because there is already a 7 in the expression, adding 2 will adjust the constant correctly.)
• “How did Han know that he could use the quadratic formula from the beginning?” (One side of the equation is a quadratic expression in the form and the other side of the equation is 0.)
• “When Han begins using the quadratic formula, why is there a positive 6 when ?” (In the quadratic formula, that term is and .)
• “Both solution strategies are correct, but the solutions look different. How can you show that they are the same values?” (The 3 part is the same, so we need to show that . Squaring those terms, and . Because both expressions are positive values that square to give the same value, .)

This activity gives students the opportunity to review solving quadratic equations. Look for students who use different strategies to compare during discussion.

The purpose of the discussion is for students to articulate methods for solving quadratic equations.

Select 2–3 previously identified pairs to share their strategies and reasons for choosing a particular solution method. If two pairs solved the same problem but got answers that look different, ask the class to verify that the solutions are equivalent. Ask students why they think one of the equations has only 1 solution rather than 2 like every other equation here. (If we graph , it touches the -axis at the vertex only.)