Previously, students saw that a squared expression of the form \((x+n)^2\) is equivalent to \(x^2 + 2nx + n^2\). This means that, when written in standard form \(ax^2 + bx +c\) (where \(a\) is 1), \(b\) is equal to \(2n\) and \(c\) is equal to \(n^2\). Here, students begin to reason the other way around. They recognize that if \(x^2 + bx +c\) is a perfect square, then the value being squared to get \(c\) is half of \(b\), or \(\left(\frac {b}{2}\right)^2\). Students use this insight to build perfect squares, which they then use to solve quadratic equations.
Students learn that if we rearrange and rewrite the expression on one side of a quadratic equation to be a perfect square, that is, if we complete the square, we can find the solutions of the equation.
Rearranging parts of an equation strategically so that it can be solved requires students to make use of structure (MP7). Maintaining the equality of an equation while transforming it prompts students to attend to precision (MP6).
Lesson overview
 12.1 Warmup: Perfect or Imperfect? (5 minutes)
 12.2 Activity: Building Perfect Squares (10 minutes)

12.3 Activity: Dipping Our Toes in Completing the Square (20 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 12.4 Cooldown: Make It a Perfect Square (5 minutes)
Learning goals:
 Comprehend that to “complete the square” is to determine the value of \(c\) that will make the expression \(x^2+bx+c\) a perfect square.
 Describe (orally and in writing) how to complete the square.
 Solve quadratic equations of the form \(x^2+bx+c=0\) by rearranging terms and completing the square.
Learning goals (student facing):
 Let’s learn a new method for solving quadratic equations.
Learning targets (student facing):
 I can explain what it means to “complete the square” and describe how to do it.
 I can solve quadratic equations by completing the square and finding square roots.
Glossary:
 completing the square  Completing the square in a quadratic expression means transforming it into the form \(a(x+p)^2q\), where \(a\), \(p\), and \(q\) are constants. Completing the square in a quadratic equation means transforming into the form \(a(x+p)^2=q\).
 Access the complete Algebra 1 Course Glossary.
Standards:
 This lesson builds towards the standards: CCSS.HSAREI.B.4.aMS.AREI.4aMO.A1.REI.A.2aMO.A1.REI.A.2b CCSS.HSAREI.B.4.bMS.AREI.4bMO.A1.REI.A.2c
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