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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.
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

  • Item Type(s): This benchmark will be assessed using: EE , GRID item(s)
  • N/A
  • 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