3.4-Miscellaneous Equations

Transcription

1 .-Miscellaneos Eqations Factoring Higher Degree Polynomials: Many higher degree polynomials can be solved by factoring. Of particlar vale is the method of factoring by groping, however all types of factoring will be sed to solve higher degree polynomial eqations. Eample: Solve the polynomial by factoring. Soltion: Factor by Groping ± ± i Therefore, the soltions are{,±i} Eample: Solve the polynomial by factoring. Soltion: Factor by Groping Therefore, the soltions are{, ± i } ± ± i

2 Eample: Solve the polynomial 6 Soltion: Solve by factoring: The factor is not factorable, se another method sch as completing the sqare. i ± ± ± Therefore, the soltions are{ },, i ± Eample: Solve the polynomial 7 Soltion: Solve by factoring: The factor is not factorable, se another method sch as completing the sqare ± i i i ± ± ± Therefore, the soltions are ±,, i

3 Eqations with Radicals: When solving an eqation containing a radical the primary objective is to isolate the term containing the radical. Once this is accomplished, raise both sides of the eqation to an appropriate power in order to eliminate the radical. Solve the remaining eqation. If the eqation still contains a radical, repeat the process. Eample: Solve the eqation Soltion: Isolate the radical and then raise both sides to the reciprocal power. Becase this is an eqation containing a sqare root, we need to check for etraneos roots. By checking both soltions in the original eqation, we determine that is not a soltion. Therefore, the only soltion is { } Eample: Solve the eqation Soltion: Isolate the radical and then raise both sides to the reciprocal power. 6 6 By checking both soltions in the original eqation, we determine that is not a soltion. Therefore, the only soltion is { 6 }

4 Eample: Solve the eqation Soltion: Isolate the radical and then raise both sides to the reciprocal power. Becase this is an eqation containing a sqare root, we need to check for etraneos roots. By checking both soltions in the original eqation, we determine that both answers check. Therefore, the soltions are{ }, Eample: Solve the eqation Soltion: Isolate the radical and then raise both sides to the reciprocal power., 6 6 By checking both soltions in the original eqation, we determine that is not a soltion. Therefore, the only soltion is { }

5 Eqations with Rational Eponents If an eqation contains a rational eponent, raise both sides of the eqation to an appropriate power to eliminate the eponent. Solve the reslting eqation. Eample: Solve the eqation 6 Soltion: Raise both sides of the eqation to a power which is the reciprocal of the power of. 6 ± 6 Eample: Solve the eqation 6 Soltion: Raise both sides of the eqation to a power which is the reciprocal of the power of

6 U-Sbstittion: The procedre called -sbstittion is a very common and etremely sefl mathematical tool. It is sed to create a qadratic eqation ot of an eqation that is not a qadratic eqation. When a polynomial is written in descending order, we generally let the variable of the middle term be eqal to another variable, sch as, hence the name -sbstittion. Eample: Solve the eqation Soltion: This eqation is not a qadratic eqation, bt rather a th degree eqation. We can create a qadratic eqation by doing a sbstittion. Let, then after sqaring both sides. Sbstitte these vales of into the eqation and solve by any acceptable method. In this case, I sed factoring. Becase or objective is to solve for not, we mst back-sbstitte, that is, replace all the -terms with -terms. ± ± The final answer contains for soltions. Check these soltions in the original eqation. Eample: Solve the eqation t t 8 Soltion: This eqation is not a qadratic eqation, bt rather an eqation with rational eponents which is really a radical eqation. We can create a qadratic eqation by doing a sbstittion. Let t, then t after sqaring both sides. Sbstitte these vales of into the eqation and solve by any acceptable method. In this case I sed factoring. t t Becase or objective is to solve for t not, we mst back-sbstitte, that is, replace all the -terms with t-terms.

7 8 8 8 t t t t t t The final answer contains two soltions. Check these soltions in the original eqation. Eample: Solve the eqation Soltion: This eqation is not a qadratic eqation, bt rather a radical eqation. We can create a qadratic eqation by doing a sbstittion. Let, then after sqaring both sides. Sbstitte these vales of into the eqation and solve by any acceptable method. In this case, I sed the sqare root property. Becase or objective is to solve for not, we mst back-sbstitte, that is, replace all the -terms with -terms. The final answer has one soltion. Check this soltion in the original eqation.

8 Eqations with Absolte Vale: Absolte vale is the distance from zero on the nmber line. However, to solve eqations containing absolte vale epressions we se the more formal definition of absolte vale. This definition will provide s with a procedre for solving absolte vale eqations., if. >, if. < This definition states that, if > and, if <. It is important to interpret as the opposite of and not negative. Eample: Solve the eqation 6 8 Soltion: Solve by sing the definition of absolte vale as discssed in a previos set of notes The soltion is therefore {,,,6} Eample: Solve the eqation Soltion: Solve by sing the definition of absolte vale as discssed in a previos set of notes. 6 The soltion is therefore {,,, }

9 .-Applications Eample: One leg of a right triangle is cm longer than the other leg. What is the length of the short leg if the total length of the hypotense and the short leg is cm? Soltion: Let the length of the short leg. Then is the length of the other leg. The hypotense will be defined as. If the hypotense pls the short leg are eqal to we will have the following eqation to solve. Since we now have a qadratic, se any appropriate method. ± 8 6 ±.66.8,.8 The only answer that makes sense in the contet of the problem is.8 which also checks. Therefore the shortest side of the triangle is.8.

10 Eample: The minimm displacement D for a yacht with length. m and a sail area of.6 m is determined by the eqation L.S.8D 6.6 Find the boats minimm displacement. Soltion: Solve the eqation for D. L.S D.8D D D D.8.8D D 6.6 The minimm displacement is.8 6.6

11 Eample: If the width of the base of a cbic container is increased by m and the length of the base decreased by m, then the volme of the new container is 6 m. What is the height of the cbic container? Soltion: Let eqal the dimensions of the original cbic container. The new width is eqal to and the new length is -. The height will be. This gives s the following eqation based on the formla for the volme of a cbic container h w l V. 6 6 V Since this is a rd degree polynomial, try factoring by groping. 6 6 Use the zero prodct property to obtain ± Therefore, the height of the cbic container is.

12 Eample: The capsize screening vale C is defined as 6 d b C Where b is the beam width in feet and d is the displacement in ponds. The Bahia 6 is a 6 ft catamaran with a capsize screening vale of. and a beam of 6. ft. Find the displacement of the Bahia 6. Soltion: Solve the eqation for d., C b d b C d d b C d b C d b C The displacement is,..

13 Eample: A tor operator ses the eqation C to find his cost in dollars for taking people on a tor of San Francisco. For what vale of is the cost $6.? If he charges $. per person for the tor, then what is his break-even point? Soltion: Solve this eqation for. C 6 6 6,6 6 6,6 Since we now have a qadratic, se any appropriate method. b ± b ac a ± 76 8 ± 6 8., The nmber. does not check in the original eqation so the only soltion is. The revene fnction will be R. To find the break-even point, set the two eqations eqal and solve. 7 Since we now have a qadratic, se any appropriate method. ± 76 8 ±. 8,. Since a negative vale does not make sense in the contet of the problem, the only soltion is. The break-even point is people.

14 Eample: Two vertical poles of lengths 6 feet and 8 feet stand feet apart. A cable reaches from the top of one pole to some point on the grond between the poles and then to the top of the other pole. Where shold this point be located to se eactly 8 feet of cable? Soltion: The soltion to this problem reqires the se of the Pythagorean Theorem. Becase there are two triangles, we will se the Pythagorean Theorem twice. For each triangle, the length of the cable is eqal to the hypotense of the triangle. For the small triangle, the hypotense is: 6 For the large triangle the hypotense is: 6 The total length of the cable is the sm of the two hypotenses. Since we are told that this length mst be 8 feet, we have the eqation: Solve this eqation: ,6 7,8 8,6 7, , ,6, ,66 6 Since both answers check in the original eqation, the point shold be located approimately 7. feet or. feet from the base of the 6 foot pole.

15 Eample: William, Nancy and Edgar met at the lodge at 8: AM, and William began hiking west at mph. At : AM., Nancy began hiking north at mph and Edgar went east on a three-wheeler at mph. At what time was the distance between Nancy and Edgar miles greater than the distance between Nancy and William? Soltion: Setting p this problem will reqire se of two basic concepts in mathematics; the Pythagorean Theorem, and the niform motion formla D RT. It will be helpfl to draw a diagram of this scenario. Let the time William begins walking. Before we can apply the Pythagorean Theorem we need to determine the length distance of each leg of the triangles. To do this we will se the motion formla. Williams Leg D RT Nancy s Leg D RT - Edgars Leg D RT - Now we can apply the Pythagorean Theorem with the given distances. D D The eqation is based on the fact that D D 8 6 6,. The vale. does not check in the original eqation, therefore the time is hors or : PM.