UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction


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1 Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the re of trpezoid, A= ( + ) h, is formul ecuse it involves two 1 or more vriles tht hve specific nd specil reltionships with the other vriles. In this cse, the formul involves four vriles: 1, se of the trpezoid;, the other se of the trpezoid; h, the height of the trpezoid; nd A, the re of the trpezoid. When you rerrnge the formul, you re simply chnging the focus of the formul. In its given form, the focus is A, the re. However, when it is necessry to find the height, h ecomes the focus of the formul nd thus is isolted using proper lgeric properties. In this lesson, you will rerrnge literl equtions (equtions tht involve two or more vriles) nd formuls with degree of. Key Concepts Literl equtions nd formuls contin equl signs. Just s in ny other eqution, we must pply proper lgeric properties to mintin lnce when chnging the focus of n eqution or formul. Tht is, if you sutrct vlue from one side of the eqution, you must do the sme to the other side, nd so on. When you chnge the focus of literl eqution or formul, you re isolting the vrile in question. To isolte vrile tht is squred, perform the inverse opertion y tking the squre root of oth sides of the eqution. When you tke the squre root of rel numer there re two solutions: one is positive nd the other is negtive. Tke (), for exmple. When you squre, the result is. The sme, however, is true for ( ). When you squre, the result is still ecuse the product of two negtives is positive. In other words, ( ) =. U5171
2 Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Since tking squre root involves finding numer tht you cn multiply y itself to result in the squre, we must tke into ccount oth the positive nd negtive. Tht is, =±. When solving for squred term of formul, the focus is importnt in determining whether to use two solutions. If the focus is ny quntity tht would never e negtive in rel life, such s distnce, time, or popultion, it is pproprite to ignore the negtive. When solving for vrile in multistep eqution, first isolte the term contining the vrile using sutrction or ddition. Then determine which opertions re pplied to the vrile, nd undo them in reverse order. Common Errors/Misconceptions forgetting to use the inverse opertions in the correct order forgetting tht there re likely two solutions (one positive nd one negtive) when solving for squred term U517
3 Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Guided Prctice Exmple 1 Solve the eqution x + y = 100 for y. 1. Isolte y. Begin y sutrcting x from oth sides. x + y = 100 y = 100 x Originl eqution Sutrct x from oth sides. y y = 100 x Tke the squre root of oth sides. =± 100 x Simplify, rememering tht the result could e positive or negtive.. Summrize your result. The formul x + y = 100 cn e rewritten s y=± 100 Exmple Solve y = 3(x 7) + 8 for x. 1. Isolte x. y = 3(x 7) + 8 y 8 = 3(x 7) Given eqution Sutrct 8 from oth sides. y 8 3 x 7 = 3 3 Divide oth sides y 3. y 8 = ( x 7 ) 3 y 8 = 3 ( x 7 ) Tke the squre root of oth sides. ± y 8 = x 7 3 7± y 8 = x 3 Add 7 to oth sides. x. U5173
4 Lesson 3: Creting Qudrtic Equtions in Two or More Vriles. Summrize your result. The eqution y = 3(x 7) + 8 solved for x is x = 7± y 8 3. Exmple 3 The formul for the re of squre is A = s, where s is the length of side of the squre. Solve the formul for s. 1. Isolte s. A = s A= Formul for the re of squre s Tke the squre root of oth sides. ± A= s. Summrize your result. In this cse, the length of squre cnnot e negtive (or it would not exist), so the negtive is disregrded. The eqution A = s solved for s is s= A. U5174
5 Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Exmple 4 y k The eqution to grph n ellipse is + = 1, where (h, is the center of the ellipse, is the numer of units right nd left of the center on the ellipse, nd is the numer of units up nd down from the center on the ellipse. Solve the formul for. y (h, x 1. Isolte. = y k + 1 Eqution of n ellipse y k = Sutrct the frction 1 from oth sides. y k 1 = ( ) Multiply oth sides y. y k = (continued) U5175
6 Lesson 3: Creting Qudrtic Equtions in Two or More Vriles ± ± = = y k = y k = = Divide ech side y. Tke the squre root of oth sides.. Summrize your result. Since is the horizontl distnce from the center of the ellipse, this is nother sitution where we disregrd the negtive. = y k The eqution + 1 solved for is = x h y k. U5176
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