Lesson 12

Completing the Square (Part 1)

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Lesson Narrative

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).


Learning Goals

Teacher Facing

  • 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.

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.

CCSS Standards

Addressing

Building Towards

Glossary Entries

  • completing the square

    Completing the square in a quadratic expression means transforming it into the form \(a(x+p)^2-q\), where \(a\), \(p\), and \(q\) are constants.

    Completing the square in a quadratic equation means transforming into the form \(a(x+p)^2=q\).

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