Quadratic Equations with Irrational Solutions

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

This video 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 (√) 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. Here, students are introduced to the plus-minus symbol (±) as a way to express both solutions. Students also briefly recall the meanings of rational and irrational numbers. They see that sometimes the solutions are expressions that involve a rational number and an irrational number—for example, x = ±√8 + 3. 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).

Mathematics (2019) Grade(s): 8 - Grade 8 Accelerated

MA19.8A.6

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

UP:MA19.8A.6

Vocabulary

  • Function
  • zero of a function
  • Roots
  • parabola
  • vertex form of a quadratic expression
  • Minimum and maximum value
  • Axis of symmetry
  • Completing the square
  • Exponential growth and decay

Knowledge

Students know:
  • The vertex form of a quadratic expression asf (x) = a(x
  • h)2 + k, where (h, k) is the vertex of the parabola.
  • Techniques for generating equivalent forms of an algebraic expression including factoring and completing the square for quadratic expressions and using properties of exponents,
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

Skills

Students are able to:
  • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures,
  • Factor quadratic expressions with leading coefficient of one
  • Complete the square in quadratic expressions.

Understanding

Students understand that:
  • An expression may be written in various equivalent forms.
  • Some forms of the expression are more beneficial for revealing key properties of the function.
Mathematics (2019) Grade(s): 8 - Grade 8 Accelerated

MA19.8A.11

Select an appropriate method to solve a quadratic equation in one variable.

UP:MA19.8A.11

Vocabulary

  • quadratic equation
  • Square root
  • Factoring
  • Completing the square
  • quadratic formula
  • Derive
  • Real numbers
  • Imaginary numbers
  • Complex numbers

Knowledge

Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.

Skills

Students are able to:
  • Take the square root of both sides of an equation.
  • Factor quadratic expressions in the form x2+bx+c where the leading coefficient is one.
  • Use the factored form to find zeros of the function.
  • Complete the square.
  • Use the quadratic formula to find solutions to quadratic equations.
  • Manipulate equations to rewrite them into other forms.

Understanding

Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.6

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

UP:MA19.A1.6

Vocabulary

  • Quadratic expression
  • Zeros
  • Complete the square
  • Roots
  • Zeros
  • Solutions
  • x-intercepts
  • Maximum value
  • Minimum value
  • Factor
  • Roots
  • Exponents
  • Equivalent form
  • Vertex form of a quadratic expression

Knowledge

Students know:
  • Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

Skills

Students are able to:
  • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
  • Factor quadratic expressions.
  • Complete the square in quadratic expressions.
  • Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.

Understanding

Students understand that:
  • Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.9

Select an appropriate method to solve a quadratic equation in one variable.

UP:MA19.A1.9

Vocabulary

  • Completing the square
  • Quadratic equations
  • Quadratic formula
  • Inspection
  • Imaginary numbers
  • Binomials
  • Trinomials

Knowledge

Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • Notational methods for expressing complex numbers.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.

Skills

Students are able to:
  • Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
  • Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
  • Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
  • Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
  • Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

Understanding

Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).

CR Resource Type

Audio/Video

Resource Provider

PBS

License Type

Custom
https://aptv.pbslearningmedia.org/resource/im20-math-ep22-715/quadratic-equatio…

Custom License Type (URL)

https://aptv.pbslearningmedia.org/help/terms-of-use/

Accessibility

Video resources: includes closed captioning or subtitles

Comments or Notes

Student task statements and practice problem worksheets accompany this resource.

Approved Date

ALSDE LOGO

This site may contain links to other websites or content belonging to or originating from third parties or links to websites and features in banners or other advertising. Such external links are not investigated, monitored, or checked for accuracy, adequacy, validity, reliability, availability, or completeness by us. WE DO NOT WARRANT, ENDORSE, GUARANTEE, OR ASSUME RESPONSIBILITY FOR THE ACCURACY OR RELIABILITY OF ANY INFORMATION OFFERED BY THIRD-PARTY WEBSITES LINKED THROUGH THIS SITE.
Alabama Learning Exchange © 2002-2023 by Alabama State Department of Education is licensed under Attribution-NonCommercial-ShareAlike 4.0 International

Design by Adaptive Theme