Rewriting Quadratic Expressions in Factored Form (Part 3): Algebra 1, Episode 17: Unit 7, Lesson 8 | Illustrative Math

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form.

Through repeated reasoning, students are able to generalize the equivalence of these two forms: (x + m)(x – m) and x² – m² (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other.

Students also consider why a difference of two squares (such as x² – 25) can be written in factored form, but a sum of two squares (such as x² + 25) cannot be, even though both are quadratic expressions with no linear term.

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-ep17-78/rewriting-quadrati…

Custom License Type (URL)

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

Accessibility

Video resources: includes closed captioning or subtitles

Approved Date

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