DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C LESSON3 . 1 Name Class Date Solving Quadratic 3.1 Solving Quadratic Equations Equations by Taking by Taking Square Roots Essential Question: What is an imaginary number, and how is it useful in solving Resource Square Roots quadratic equations? Locker Common Core Math Standards Explore Investigating Ways of Solving Simple The student is expected to: Quadratic Equations COMMON CORE N-CN.A.1 There are many ways to solve a quadratic equation. Here, you will use three methods to solve the 2 Know there is a complex number i such that i 2 = –1, and every complex equation x = 16: by graphing, by factoring, and by taking square roots. number has the form a + bi with a and b real. Also A-REI.B.4b Solve x 2 16 by graphing. y A = 16 Mathematical Practices First treat each side of the equation as a function, and 14 2 12 COMMON graph the two functions, which in this case are ƒ( x) = x CORE MP.4 Modeling and g x 16 , on the same coordinate plane. 10 ( ) = 8 Language Objective Then identify the x-coordinates of the points where the 6 two graphs intersect. 4 Have students decide whether a given square root is an imaginary 2 x x = -4 or x = 4 0 number (square root of a negative number) or a real number and explain -8 -4 4 8 their reasoning to a partner. 2 B Solve x = 16 by factoring. This method involves rewriting the equation so that 0 is on one side in order to use the zero-product property, which says that the product of two numbers is 0 if and only if at least one of the numbers is 0. ENGAGE 2 Write the equation. x = 16 2 Essential Question: What is an Subtract 16 from both sides. x - 16 = 0 imaginary number, and how is it useful Factor the difference of two squares. x + 4 ( x - 4) = 0 in solving quadratic equations? ( ) Apply the zero-product property. x + 4 = 0 or x - 4 = 0 Possible answer: An imaginary number has the form Solve for x. x 4 or x 4 bi; b is a nonzero real number and i is the imaginary = - = Solve x 2 16 by taking square roots. unit, which is defined to be equal to √――- 1 . C = 2 A real number x is a square root of a nonnegative real number a provided x a . A square root is written Imaginary numbers allow you to solve quadratic _ = _ using the radical symbol √ . Every positive real number a has both a positive square root, written √ a , equations of the form x 2 a when a is a negative Mifflin Houghton © Company Harcourt Publishing _ _ = and a negative square root, written - √ a . For instance, the square roots of 9 are ± √ 9 (read_ “plus or number. minus the square root of 9”), or ±3. The number 0 has only itself as its square root: ± √ 0 = 0 . 2 Write the equation. x = 16 _ Use the definition of square root. x = ± √ 16 PREVIEW: LESSON PERFORMANCE TASK Simplify the square roots. x = ±4 Module 3 through “File info” 113 Lesson 1 DO NOT EDIT--Changes must be made CorrectionKey=NL-C;CA-C Date Class View the Engage section online. Discuss the photo Name 3 . 1 Solving Quadratic Equations Resource by Taking Square Rootsful in solving Locker What is an imaginary number, and how is it use HARDCOVER PAGES 8390 Essential Question: quadratic equations? 2 and how a quadratic function can be used to model a i = -1, and every complex number has .B.4b COMMON CORE N-CN.A.1 Know there is a complex number i such that the form a + bi with a and b real. Also A-REI Investigating Ways of Solving Simple e Explore Quadratic Equations A2_MNLESE385894_U2M03L1 113 equation. Here, you will roots.use three methods to solve th 6/8/15 2:19 PM y 16 suspension bridge. Then preview the Lesson There are many2 ways to solve a quadratic x 16: by graphing, by factoring, and by taking square equation = 14 12 2 16 by graphing. Solve x = x2 10 Turn to these pages to the equation as a function, xand) = his case are ƒ( 8 First treat each side of te plane. 6 graph the two functions, which in t 4 16 , on the same coordina x and g( x) = 2 coordinates of the points where the Performance Task. 0 4 8 4 Then identify the x- -8 - find this lesson in the two graphs intersect. 4 4 or x = -product x = - s is 0. ne side in order to use the zero 2 x = 16 by factoring. d only if at least one of the number Solve ewriting the equation so that 0 is on o product of two numbers is 0 if an hardcover student 2 This method involves r x = 16 property, which says that the 0 2 16 = Write the equation. x - 0 4 ( x - 4) = Subtract 16 from both sides. x + 0 edition. ) x 4 = ( or - 4 0 Factor the difference of two squares. x + = property. x 4 or = 4 Apply the zero-product x = - x. _ 2 a . A square root is writtena , Solve for x = t, written √ a provided _ 2 16 by taking square roots. 9 (read “plus or Solve x = of a nonnegative real numbera has both a positive square√ roo_ quare_ root ots of 9 are ± √0 = 0 . √ . Every positive_ real number A real number x is a s √a. For instance, the squareelf roas its square root: ± using the radical symbol ritten - © Houghton Mifflin Harcourt Publishing Company Publishing Harcourt Mifflin Houghton © and a negative square root, w or ±3 . The number 0 has only its 2 minus the square root of 9”), x = 16 _ √16 Write the equation. x = ± Lesson 1 4 Use the definition of square root. x = ± 113 Simplify the square roots. 6/8/15 2:19 PM Module 3 3L1 113 A2_MNLESE385894_U2M0 113 Lesson 3 . 1 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A Reflect 2 1. Which of the three methods would you use to solve x = 5? Explain, and then use the method to find the EXPLORE solutions. The graphing method would give only approximate solutions, while the factoring method Investigating Ways of Solving Simple can’t be used because x 2 5 isn’t a difference of two squares. However, taking square - Quadratic Equations roots gives x 5 . = ±√― 2 2. Can the equation x = -9 be solved by any of the three methods? Explain. None of the three methods can be used. Attempting to use the graphing method results in INTEGRATE MATHEMATICAL a parabola and a line that don’t intersect. Attempting to use the factoring method results PRACTICES 2 in the expression x + 9, which isn’t factorable. Attempting to use square roots doesn’t Focus on Critical Thinking make sense because square roots of negative numbers aren’t defined. MP.3 Discuss whether 1 can be used instead of 0 in the Zero Product Property to make a “One” Product Explain 1 Finding Real Solutions of Simple Property. Have students show that if ab = 1, neither a Quadratic Equations nor b must equal 1. Have them consider –1 and other 2 When solving a quadratic equation of the form a x + c = 0 by taking square roots, you may need to use the numbers and lead them to conclude that the property following properties of square roots to simplify the solutions. (In a later lesson, these properties are stated in a more general form and then proved.) holds only for 0. Property Name Words Symbols Numbers _ _ QUESTIONING STRATEGIES _ _ _ Product property The square root of a ab a b where a ≥ 0 √ = √ ⋅ √ √ 12 √ 4 3 of square roots product equals the product and b ≥ 0 = _ ⋅ _ How can you use the factored form of a = √ 4 ⋅ √3 of the square roots of the _ quadratic equation to find the zeros of the factors. = 2√ 3 _ related quadratic function? Factor the equation, a a Quotient The square root of a _ _√_ where a ≥ 0 ― b = _5 _ √5 property of fraction equals the quotient √ b ― ― apply the Zero Product Property, and then solve the and b > 0 9 = 9 square roots of the square roots of √― equation. the numerator and the 5 Mifflin Houghton © Company Harcourt Publishing _√― denominator. = 3 What are the roots of an equation? the values of the variable that make the equation true Using the quotient property of square roots may require an additional step of rationalizing the denominator if _2 _ √―2 the denominator is not a rational number. For instance, the quotient property allows you to write ― as , 7 √7 ― _ √―2 _ √―7 but √―7 is not a rational number. To rationalize the denominator, multiply by (a form of 1) and get EXPLAIN 1 √7 √7 2 7 14 14 ― ― this result: _√― _ √― _ √― _ √― . ⋅ = = 7 √―7 √―7 √―49 Finding Real Solutions of Simple Quadratic Equations INTEGRATE TECHNOLOGY When zeros occur at non-integer x-values, Module 3 114 Lesson 1 using a table to find them can be difficult.