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Solving Quadratic Equations With the Zero Product Property: Algebra 1, Episode 13: Unit 7, Lesson 4 | Illustrative Math

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Title:

 Solving Quadratic Equations With the Zero Product Property: Algebra 1, Episode 13: Unit 7, Lesson 4 | Illustrative Math

URL:

 https://aptv.pbslearningmedia.org/resource/im20-math-ep13-74/solving-quadratic-equations-with-the-zero-product-property/

Content Source:

 PBS
Type: Audio/Video

Overview:

In this video lesson, students learn about the zero product property. They use it to reason about the solutions to quadratic equations that each have a quadratic expression in the factored form on one side and 0 on the other side. They see that when an expression is a product of two or more factors and that product is 0, one of the factors must be 0. Students make use of the structure of a quadratic expression in factored form and the zero product property to understand the connections between the numbers in the form and the x-intercepts of its graph (MP7).

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra I
17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.A.HS.17- Solve an equation of the form ax + b = c where a, b, and c are positive whole numbers and the solution, x, is a positive whole number to represent a real-world problem.
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra I
32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth and decay.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.F.HS.32- Identify the y-intercept of a linear equation in the form of y=mx+b as (0,b).
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II
4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II
20 ) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [A-CED1]
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II with Trigonometry
4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II with Trigonometry
20 ) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [A-CED1]
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
18. Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
[Grade 7, 9, and linear portion of Algebra I with Probability, 11]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Write and solve mathematical equations (or inequalities) to model real-world problems.
  • Interpret the solution to an equation in the context of a problem.
  • Interpret the solution set of an inequality in the context of a problem.
Teacher Vocabulary:
  • Algebraic expressions
  • Equations
  • Inequalities
  • Greater than
  • Greater than or equal to
  • less than
  • less than or equal to
Knowledge:
Students know:
  • p(x + q) = px + pq, where p and q are specific rational numbers.
  • When multiplying or dividing both sides of an inequality by a negative number, every term must change signs and the inequality symbol reversed.
  • In the graph of an inequality, the endpoint will be a closed circle indicating the number is included in the solution set (≤ or ≥) or an open circle indicating the number is not included in the solution set ( < or >).
Skills:
Students are able to:
  • Use variables to represent quantities in a real-world or mathematical problem.
  • Construct equations (px + q = r and p(x + q) = r) to solve problems by reasoning about the quantities.
  • Construct simple inequalities (px + q > r or px + q < r) to solve problems by reasoning about the quantities.
  • Graph the solution set of an inequality.
Understanding:
Students understand that:
  • Real-world problems can be represented through algebraic expressions, equations, and inequalities.
  • The inequality symbol reverses when multiplying or dividing both sides of an inequality by a negative number, and why.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems.
  • Produce the useful equivalent forms of expressions,
  • Factor a quadratic expression with leading coefficient of one to reveal the zeros of the function it defines and complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines.
  • Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation.
Teacher 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.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.6.1: Convert an expression to an alternative format.
ALGI.6.2: Recognize the best format for a specific application.
ALGI.6.3: Match equivalent expressions written in different forms.

a.
ALGI.6.4: Define factor, quadratic expression and zero product property.
ALGI.6.5: Factor a quadratic expression.
ALGI.6.6: Use the zero product property to reveal the zeros in the function.
ALGI.6.7: Solve a one-step equation.
ALGI.6.8: Solve a two-step equation.
ALGI.6.9: Determine the Greatest Common Factor (GCF).

b.
ALGI.6.10: Define maximum and minimum value.
ALGI.6.11: Explain the steps for completing the square.
ALGI.6.12: Given a quadratic expression in which the square has already been completed, determine the maximum or minimum values.

c.
ALGI.6.13: Define roots.
ALGI.6.14: Find the equation using the distributive property.
ALGI.6.15: Locate and identify the roots on a graph using the x-intercepts.
ALGI.6.16: Take given roots and convert into a one-step equation set equal to zero.

Prior Knowledge Skills:
  • Identify how many solutions the linear equation may or may not have.
  • Recall how to solve problems using the distributive property
  • Explain the distributive property.
  • Recall solving one-step equations.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division.

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
9. Select an appropriate method to solve a quadratic equation in one variable.

a. 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. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Solve quadratic equations where both sides of the equation have evident square roots by inspection.
  • Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square.
  • Use the method of completing the square on the equation in standard form ax2+bx+c=0 to derive the quadratic formula.
  • Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation.
  • Use the quadratic formula to solve quadratic equations.
  • Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi.
  • Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true.
Teacher 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).
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.9.1: Define quadratic equation and zero product property.
ALGI.9.2: Solve one-step equations using addition and subtraction that are set equal to zero.
ALGI.9.3: Solve two-step equations using addition and subtraction that are set equal to zero.

a.
ALGI.9.4: Define completing the square.
ALGI.9.5: 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.
ALGI.9.6: Derive the quadratic formula from the form (x - p)= q.

b.
ALGI.9.7: Define quadratic formula, factoring, square root, complex number, and real number.
ALGI.9.8: Solve quadratic equations by completing the square.
ALGI.9.9: Solve quadratic equations by the quadratic formula.
ALGI.9.10: Solve quadratic equations by factoring.
ALGI.9.11: Solve quadratic equations by taking square roots.
ALGI.9.12: Recognize when the quadratic formula gives complex solutions.
ALGI.9.13: Write complex solutions as a ±bi for real numbers a and b.

Prior Knowledge Skills:
  • Identify perfect squares and square roots.
  • Define square root, expressions, and approximations.
  • Explain the distributive property.
  • Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
  • Recalving one-step equations.
  • List given information from the problem.
  • Identify the unknown, in a given situation, as the variable.
  • Test the found number for accuracy by substitution.
    Example: Is 5 an accurate solution of 2(x + 5)=12?
  • Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
  • Define equation and variable.
  • Set up an equation to represent the given situation, using correct mathematical operations and variables.
  • Recognize the correct order to solve expressions with more than one operation.
  • Calculate a numerical expression (Ex. V=4x4x4).
  • Choose the correct value to replace each variable in the algebraic expression (Substitution).
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.9 Identify equivalent expressions given a linear expression using arithmetic operations.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable.
  • Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.
Teacher Vocabulary:
  • Variable
  • Equation
  • Inequality
  • Solution Set
  • Identity
  • No solution for a given domain
  • Approximate solutions
Knowledge:
Students know:
  • When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or rational functional relationship.
Skills:
Students are able to:
  • Write equations in one variable that accurately model contextual situations.
Understanding:
Students understand that:
  • Features of a contextual problem can be used to create a mathematical model for that problem.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.11.1: Solve the equation represented by the real-world situation.
ALGI.11.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.11.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.11.4: Define equation, expression, variable, equality and inequality.
ALGI.11.5: Create inequalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.6: Create equalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.7: Solve two-step equations and inequalities.
ALGI.11.8: Solve one-step equations and inequalities using the four basic operations.
ALGI.11.9: Compare and contrast equations and inequalities.
ALGI.11.10: Recognize inequality symbols including greater than, less than, greater than equal to and less than equal to.

Prior Knowledge Skills:
  • Test the found number or number set for accuracy by substitution.
  • Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
  • Define equation, inequality, and variable.
  • Convert mathematical terms to mathematical symbols and numbers.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.11 Select an equation or inequality involving one operation (limit to addition or subtraction) with one variable that represents a real-world problem. Solve the equation.
Tags: coefficients, complex solutions, equation, equivalent, factor, function, inequalities, linear equation, quadratic, variable
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Author: Hannah Bradley
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