MAFS.912.A-REI.2.4

Solve quadratic equations in one variable.
  1. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
  2. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Algebra: Reasoning with Equations & Inequalities
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Solve equations and inequalities in one variable. (Algebra 1 - Major Cluster) (Algebra 2 - Supporting Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

  • Assessment Limits :
    In items that require the student to transform a quadratic equation to
    vertex form, b/a must be an even integer.

    In items that require the student to solve a simple quadratic equation
    by inspection or by taking square roots, equations should be in the
    form ax² = c or ax² + d = c, where a, c, and d are rational numbers and
    where c is not an integer that is a perfect square and c – d is not an
    integer that is a perfect square.

    In items that allow the student to choose the method for solving a
    quadratic equation, equations should be in the form ax² + bx + c = d,
    where a, b, c, and d are integers.

    Items may require the student to recognize that a solution is nonreal
    but should not require the student to find a nonreal solution. 

  • Calculator :

    Neutral

  • Clarification :
    Students will rewrite a quadratic equation in vertex form by completing the square. 

    Students will use the vertex form of a quadratic equation to complete steps in the derivation of the quadratic formula. 

    Students will solve a simple quadratic equation by inspection or by taking square roots. 

    Students will solve a quadratic equation by choosing an appropriate method (i.e., completing the square, the quadratic formula, or factoring). 

    Students will validate why taking the square root of both sides when solving a quadratic equation will yield two solutions. 

    Students will recognize that the quadratic formula can be used to find complex solutions.

  • Stimulus Attributes :
    The formula must be given in the item for items that can only be solved using the quadratic formula.

    Items should be set in a mathematical context.

    Items may use function notation.

     

  • Response Attributes :
    Items may require the student to complete a missing step in the
    derivation of the quadratic formula.

    Items may require the student to provide an answer in the form
    (x – p)² = q.

    Items may require the student to recognize equivalent solutions to
    the quadratic equation.

    Responses with square roots should require the student to rewrite
    the square root so that the radicand has no square factors.

Sample Test Items (2)
  • Test Item #: Sample Item 1
  • Question:

    Matthew solved the quadratic equation shown.

    4x²-24x+7=3

    One of the steps that Matthew used to solve the equation is shown.

    Drag the values into the boxes to complete the step and the solution.

     

  • Difficulty: N/A
  • Type: GRID: Graphic Response Item Display

  • Test Item #: Sample Item 2
  • Question:

    An equation is shown.

    3x²+7x=1

    The formula begin mathsize 12px style x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end style can be used to solve the equation.

    Click on the blank to enter a numeric expression that is one solution to the given equation.

  • Difficulty: N/A
  • Type: EE: Equation Editor

Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1207310: Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1206330: Analytic Geometry (Specifically in versions: 2014 - 2015 (course terminated))
1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200335: Algebra 2 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2019 (course terminated))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.912.A-REI.2.AP.4a: Solve quadratic equations by completing the square.
MAFS.912.A-REI.2.AP.4b: Solve quadratic equations by using the quadratic formula.
MAFS.912.A-REI.2.AP.4c: Solve quadratic equations by factoring.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Assessments

Sample 4 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 3 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 1 - High School Algebra 1 State Interim Assessment:

This is the State Interim Assessment for high school.

Type: Assessment

Educational Games

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

Formative Assessments

Quadratic Formula - 1:

Students are asked to derive the quadratic formula by completing the square.

Type: Formative Assessment

Quadratic Formula - 2:

Students are asked to complete the derivation of the quadratic formula.

Type: Formative Assessment

Complete the Square - 1:

Students are asked to solve a quadratic equation by completing the square.

Type: Formative Assessment

Complete the Square - 2:

Students are asked to solve a quadratic equation by completing the square.

Type: Formative Assessment

Complete the Square - 3:

Students are asked to solve a quadratic equation by completing the square.

Type: Formative Assessment

Complex Solutions?:

Students are asked to explain how to recognize when the quadratic formula results in complex solutions.

Type: Formative Assessment

Which Strategy?:

Students are shown four quadratic equations and asked to choose the best method for solving each equation.

Type: Formative Assessment

Lesson Plans

Solving Quadratic Equations by Completing the square:

Students will model the process of completing the square (leading coefficient of 1) with algebra tiles, then practice solving equations after using the completing the square method. This lesson provides a discovery opportunity to conceptually see why the process of squaring half of the b value is considered completing the square.

Type: Lesson Plan

Solving Quadratics - Exploring Different Methods:

Students will explore how different methods find the solutions (roots) to quadratic equations including, factoring, graphing, and the quadratic formula.

Type: Lesson Plan

Ranking Sports Players (Quadratic Equations Practice):

Students will rank sports players designing methods, using different indicators, and working with quadratic equations.

Type: Lesson Plan

Sorting Equations and Identities:

This lesson is intended to help you assess how well students are able to:

  • Recognize the differences between equations and identities.
  • Substitute numbers into algebraic statements in order to test their validity in special cases.
  • Resist common errors when manipulating expressions such as 2(x – 3) = 2x – 3; (x + 3)2 = x2 + 32.
  • Carry out correct algebraic manipulations.
It also aims to encourage discussion on some common misconceptions about algebra.

Type: Lesson Plan

Solving Quadratic Equations: Cutting Corners:

This lesson unit is intended to help you assess how well students are able to solve quadratics in one variable. In particular, the lesson will help you identify and help students who have the following difficulties; making sense of a real life situation and deciding on the math to apply to the problem, solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring, and interpreting results in the context of a real life situation.

Type: Lesson Plan

The Quadratic Quandary:

Students will sort various quadratic equations by the method they would use for solving (ie. factoring, quadratic formula). Then as a class they justify their placements and eventually discover that there are many ways to solve and that some make sense in different situations, however there is no real "correct" method for each equation type.

Type: Lesson Plan

Original Student Tutorials

Solving Rational Equations: Using Common Denominators:

Learn how to solve rational functions by getting common denominators in this interactive tutorial.

Type: Original Student Tutorial

Solving Rational Equations: Cross Multiplying:

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Type: Original Student Tutorial

Problem-Solving Tasks

Braking Distance:

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

Type: Problem-Solving Task

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

Springboard Dive:

The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Type: Problem-Solving Task

Tutorials

Solving Quadratic Equations Using the Quadratic Formula:

You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.

Type: Tutorial

Learning How to Complete the Square:

You will learn int his video how to solve the Quadratic Equation by Completing the Square.

Type: Tutorial

Solving Quadratic Equations by Square Roots:

In this video tutorial students will learn how to solve quadratic equations by square roots.

Type: Tutorial

Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

 
 
 

Using this CMAP

To view an introduction on the CMAP tool, please click here

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx 

Type: Unit/Lesson Sequence

Virtual Manipulative

Solving Quadratics By Taking The Square Root:

This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.

Type: Virtual Manipulative

Worksheet

Three Basic Methods for Solving Quadratic Equations:

The activity/resources reviews methods for solving quadratic equations including factoring, using the quadratic formula, and completing the square.

Type: Worksheet

STEM Lessons - Model Eliciting Activity

Ranking Sports Players (Quadratic Equations Practice):

Students will rank sports players designing methods, using different indicators, and working with quadratic equations.

MFAS Formative Assessments

Complete the Square - 1:

Students are asked to solve a quadratic equation by completing the square.

Complete the Square - 2:

Students are asked to solve a quadratic equation by completing the square.

Complete the Square - 3:

Students are asked to solve a quadratic equation by completing the square.

Complex Solutions?:

Students are asked to explain how to recognize when the quadratic formula results in complex solutions.

Quadratic Formula - 1:

Students are asked to derive the quadratic formula by completing the square.

Quadratic Formula - 2:

Students are asked to complete the derivation of the quadratic formula.

Which Strategy?:

Students are shown four quadratic equations and asked to choose the best method for solving each equation.

Original Student Tutorials Mathematics - Grades 9-12

Solving Rational Equations: Cross Multiplying:

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Solving Rational Equations: Using Common Denominators:

Learn how to solve rational functions by getting common denominators in this interactive tutorial.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Original Student Tutorials

Solving Rational Equations: Using Common Denominators:

Learn how to solve rational functions by getting common denominators in this interactive tutorial.

Type: Original Student Tutorial

Solving Rational Equations: Cross Multiplying:

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Type: Original Student Tutorial

Educational Games

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

Problem-Solving Tasks

Braking Distance:

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

Type: Problem-Solving Task

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

Springboard Dive:

The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Type: Problem-Solving Task

Tutorials

Solving Quadratic Equations Using the Quadratic Formula:

You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.

Type: Tutorial

Learning How to Complete the Square:

You will learn int his video how to solve the Quadratic Equation by Completing the Square.

Type: Tutorial

Solving Quadratic Equations by Square Roots:

In this video tutorial students will learn how to solve quadratic equations by square roots.

Type: Tutorial

Virtual Manipulative

Solving Quadratics By Taking The Square Root:

This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.

Type: Virtual Manipulative

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Tasks

Braking Distance:

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

Type: Problem-Solving Task

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

Type: Problem-Solving Task

Springboard Dive:

The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

Type: Problem-Solving Task

Virtual Manipulative

Solving Quadratics By Taking The Square Root:

This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.

Type: Virtual Manipulative