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Applying the Quadratic Formula (Part 1): Algebra 1, Episode 24: Unit 7, Lesson 17 | Illustrative Math

Classroom Resource Information

Title:

Applying the Quadratic Formula (Part 1): Algebra 1, Episode 24: Unit 7, Lesson 17 | Illustrative Math

URL:

https://aptv.pbslearningmedia.org/resource/im20-math-ep24-717/applying-the-quadratic-formula-part-1/

Content Source:

PBS

Type:

Audio/Video

Overview:

In this video lesson, students return to some quadratic functions they have seen. They write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward (MP5).

The work in this lesson—writing equations, solving them, and interpreting the solutions in context—encourages students to reason quantitatively and abstractly (MP2).

Content Standard(s):

Mathematics MA2015 (2016) Grade: 9-12 Algebra I	16 ) Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1]
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 II	4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]
Mathematics MA2015 (2016) Grade: 9-12 Algebra II	13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).
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	24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]
Mathematics MA2015 (2016) Grade: 9-12 Algebra II	29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5] Example: If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
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	13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).
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 MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry	24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]
Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry	29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5] Example: If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
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	5. Use the structure of an expression to identify ways to rewrite it. Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²). 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 and more efficient ways. Teacher Vocabulary: Terms Linear expressions Equivalent expressions Difference of two squares Factor Difference of squares Knowledge: Students know: Algebraic properties. When one form of an algebraic expression is more useful than an equivalent form of that same expression. Skills: Students are able to: Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures. Understanding: Students understand that: Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions. Diverse Learning Needs: Essential Skills: Learning Objectives: ALGI.5.1: Define equivalent expressions. ALGI.5.2: Rewrite an exponential expression in an alternative way. ALGI.5.3: Rewrite a quadratic expression in an alternative way. ALGI.5.4: Rewrite a linear expression in an alternative form. ALGI.5.5: Understand that rewriting an expression in different forms in a problem context can shed light on the problem. ALGI.5.6: Recall properties of exponents. Prior Knowledge Skills: li>Give examples of the properties of operations including distributive, commutative, and associative. Recall how to find the greatest common factor. Combine like terms of a given expression. Recognize the property demonstrated in a given expression. Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x). Simplify an expression by dividing by the greatest common factor (Ex. 18x + 6y= 6(3x + y). Define linear expression, rational, coefficient, and rational coefficient. Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). 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	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.
Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability	15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions. Unpacked Content Evidence Of Student Attainment: Students: Given input/output relations between two variables in graphical form, verbal description, set of ordered pairs, or algebraic model, distinguish between those that are functions and non-functions. Using functional notation, Evaluate functions for inputs. Interpret statements in terms of context. Given a contextual relationship that may be represented as a function, Determine that exactly one element of the range (output) is assigned to each element of the domain (input) by the function. Relate the domain to its graph and to the quantitative relationship it describes. Teacher Vocabulary: Domain Range Function Relation Function notation Set notation Knowledge: Students know: Distinguishing characteristics of functions. Conventions of function notation. In graphing functions the ordered pairs are (x,f(x)) and the graph is y = f(x). Skills: Students are able to: Evaluate functions for inputs in their domains. Interpret statements that use function notation in terms of context. Accurately graph functions when given function notation. Accurately determine domain and range values from function notation. Understanding: Students understand that: A function is a mapping of the domain to the rangeFunction notation is useful in contextual situations to see the relationship between two variables when the unique output for each input relation is satisfied. Diverse Learning Needs: Essential Skills: Learning Objectives: ALGI.15.1: Define domain, range, relation, function, table of values, input, and output. ALGI.15.2: Understand the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. ALGI.15.3: Understand that a function is a rule that assigns to each input exactly one output. ALGI.15.4: Identify the equation of a function, given its graph. ALGI.15.5: Find the range of a function given its domain. ALGI.15.6: Recognize that f(x) and y are the same. ALGI.15.7: Recall how to complete input/output tables. ALGI.15.8: Recall how to read/interpret information from a table. ALGI.15.9: Define function notation. ALGI.15.10: Translate a simple word problem into function notation. ALGI.15.11: Evaluate function when given x-value. Prior Knowledge Skills: Analyze the graph to determine the rate of change. Generate the slope of a line using given ordered pairs. Define linear functions, nonlinear functions, slope, and y-intercept Identify ordered pairs. Plot points on a coordinate plane., then connect points for the vertices to sketch a polygon. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.12.15 Use the vertical line test to determine if a given relation is a function.

Tags:

coefficients, complex solutions, domain, equations, function, graph, inequalities, linear equations, quadratic equations, radical equations, rational equations, variable

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Hannah Bradley

ALEX Classroom Resource

Applying the Quadratic Formula (Part 1): Algebra 1, Episode 24: Unit 7, Lesson 17 | Illustrative Math