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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §8.4 Eqns w/ Quadratic Form

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review § Any QUESTIONS About §8.3 → Quadratic Fcn Applications Any QUESTIONS About HomeWork §8.3 → HW-39 8.3 MTH 55

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 3 Bruce Mayer, PE Chabot College Mathematics Recognizing Eqns in Quadratic Form Certain equations that are not really quadratic can be thought of in such a way that they can be solved as quadratic. For example, because the square of x 2 is x 4, the equation x 4 − 5x 2 + 4 = 0 is said to be “quadratic in x 2 ”

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 4 Bruce Mayer, PE Chabot College Mathematics Quadratic Form Continuing from the previous Slide x 4 – 5x 2 + 4 = 0 (x 2 ) 2 – 5(x 2 ) + 4 = 0 u 2 – 5u + 4 = 0. The last equation can be solved by factoring or by the quadratic formula. Then, remembering that u = x 2, we can solve for x Equations that can be solved like this are reducible to, or are in, quadratic form.

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example Solve x 4 – 5x 2 + 4 = 0 SOLUTION Let u = x 2. Then we solve by substituting u for x 2 and u 2 for x 4 : u 2 – 5u + 4 = 0 (u – 1)(u – 4) = 0 u = 1 or u = 4 u – 1 = 0 or u – 4 = 0 Factoring Principle of zero products

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example Solve x 4 – 5x 2 + 4 = 0 Recall u = x 2 x 2 = 1 or x 2 = 4 Replace u with x 2 To check, note that for both x = 1 and x = −1, we have x 2 = 1 and x 4 = 1. Similarly, for both x = 2 and x = −2, we have x 2 = 4 and x 4 = 16. Thus instead of making four checks, we need make only two.

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example Solve x 4 – 5x 2 + 4 = 0 CHECK: x = 1:x = 2: x 4 – 5x 2 + 4 = 0 (1) – 5(1) + 4 = 0 x 4 – 5x 2 + 4 = 0 (16) – 5(4) + 4 = 0 TRUE STATE: The solutions are 1, −1, 2, and −2.

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 8 Bruce Mayer, PE Chabot College Mathematics Quadratic Form Quandry Caution! A common error on problems like the previous example is to solve for u but forget to solve for x. Remember to solve for the original variable!

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 9 Bruce Mayer, PE Chabot College Mathematics Radical and Rational Equations Sometimes rational equations, radical equations, or equations containing exponents that are fractions are reducible to quadratic. It is especially important that answers to these equations be CHECKED in the original equation.

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example Solve SOLUTION: Let Next Substitute: u 2 – 8u – 9 = 0 (u – 9)(u +1) = 0 u = 9 or u = –1 u – 9 = 0 or u + 1 = 0

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example Solve Then by the Principle of Zero Products CHECK x = 81:x = 1: FALSE TRUE

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 12 Bruce Mayer, PE Chabot College Mathematics To Solve an Equation That is Reducible to Quadratic Form 1.The equation is quadratic in form if the variable factor in one term is the square of the variable factor in the other variable term. 2.Note Carefully any substitutions made

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 13 Bruce Mayer, PE Chabot College Mathematics To Solve an Equation That is Reducible to Quadratic Form 3.Whenever you making a substitution, remember to solve for the variable that is used in the original equation. 4.Check possible answers in the ORIGINAL equation.

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 14 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work Problems From §8.4 Exercise Set 6, 12, 18, 24, 28, 48 QUADRATIC IN FORM Lyngen kirke (Lyngen Church) A steeple was added to the west side of the church during the reconstruction period in 1840-46. It is a majestic steeple with a NeoGothic character, and is approximately QUADRATIC IN FORM.

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 15 Bruce Mayer, PE Chabot College Mathematics All Done for Today Prob48 “Cleans Up”

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 16 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 17 Bruce Mayer, PE Chabot College Mathematics Graph y = |x| Make T-table

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BMayer@ChabotCollege.edu MTH55_Lec-53_Fa08_sec_8-4_Eqns_Quadratic_in_Form.ppt 18 Bruce Mayer, PE Chabot College Mathematics

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