Completing the Square (Part 2)

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

In this video lesson, students learn that completing the square can be used to solve any quadratic equation, including equations that involve rational numbers that are not integers. Students notice that the process of completing the square is the same when the equations involve messier numbers as when they have simple integers, but the calculations may be more time consuming and prone to error. An error-analysis activity highlights some common errors related to completing the square.

Completing the square for quadratic expressions that are more elaborate encourages students to look for and make use of the same structure that helped them when they were working with less complicated expressions (MP7).

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

MA19.8A.11
Select an appropriate method to solve a quadratic equation in one variable.

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.A1.9
Select an appropriate method to solve a quadratic equation in one variable.

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-ep21-713/completing-the-sq…

Custom License Type (URL)

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

Accessibility

Video resources: includes closed captioning or subtitles

Comments or Notes

There are student task statements and practice problem worksheets that accompany this resource. There are a Part One and Part Three videos for this resource.

Approved Date

2022-08-17

Design by Adaptive Theme