Lesson 15

Quadratic Equations with Irrational Solutions

Lesson Narrative

This lesson serves two main purposes: to reiterate that some solutions to quadratic equations are irrational, and to give students the tools to express those solutions exactly and succinctly.

Students recall that the radical symbol (\(\sqrt{\phantom{3}}\)) can be used to denote the positive square root of a number. Many quadratic equations have a positive and a negative solution, and up until this point, students have been writing them separately. For example, the solutions of \(x^2=49\) are \(x=7\) and \(x=\text-7\). Here, students are introduced to the plus-minus symbol (\(\pm\)) as a way to express both solutions (for example, \(x=\pm7\)).

Students also briefly recall the meanings of rational and irrational numbers. (They will have a more thorough review later in the unit.) They see that sometimes the solutions are expressions that involve a rational number and an irrational number—for example, \(x=\pm \sqrt8+3\). While this is a compact, exact, and efficient way to express irrational solutions, it is not always easy to intuit the size of the solutions just by looking at the expressions. Students make sense of these solutions by finding their decimal approximations and by solving the equations by graphing. The work here gives students opportunities to reason quantitatively and abstractly (MP2).

Learning Goals

Teacher Facing

  • Coordinate and compare (orally and in writing) solutions to quadratic equations obtained by completing the square and those obtained by graphing.
  • Understand that the “plus-minus” symbol is used to represent both square roots of a number and that the square root notation expresses only the positive square root.
  • Use radical and “plus-minus” symbols to express solutions to quadratic equations.

Student Facing

  • Let’s find exact solutions to quadratic equations even if the solutions are irrational.

Required Materials

Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Learning Targets

Student Facing

  • I can use the radical and “plus-minus” symbols to represent solutions to quadratic equations.
  • I know why the plus-minus symbol is used when solving quadratic equations by finding square roots.

CCSS Standards

Building On


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 \(a(x+p)^2=q\).

  • irrational number

    An irrational number is a number that is not rational. That is, it cannot be expressed as a positive or negative fraction, or zero.

  • perfect square

    A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.

  • rational number

    A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into \(b\) equal parts and finding the point that is \(a\) of them from 0. We can always write a fraction in the form \(\frac{a}{b}\) where \(a\) and \(b\) are whole numbers, with \(b\) not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as \(\frac{7}{10}\).

    The numbers \(3\), \(\text-\frac34\), and \(6.7\) are all rational numbers. The numbers \(\pi\) and \(\text-\sqrt{2}\) are not rational numbers, because they cannot be written as fractions or their opposites.

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