Solving Radical Equations

Transcription

1 Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is the vlue o the vrible tht stisies the rdil eqution, tht is, mkes the rdil eqution true. Emple: Is the vlue solution to the rdil eqution? Solution: Substitute the vlue o into the eqution nd evlute. Sine the eqution results in n identity, is solution to the eqution. Solving dil Eqution: When solving n eqution ontining rdil, the primry objetive is to isolte the term ontining the rdil. One this is omplished, rise both sides o the eqution to power tht is equl to the inde o the rdil in the eqution. Emple: Solve the eqution. Solution: Isolte the rdil, nd then rise both sides o the eqution to the nd power. ) We n hek the solution by substituting the solution into the originl eqution.

2 Etrneous Solutions: When rising both sides o n eqution to the sme power, the resulting eqution my hve one or more solutions tht do not stisy the originl eqution, tht is, etrneous solutions. So when solving rdil eqution, it is very importnt to hek ll solutions in the originl eqution. Emple: Solve the eqution Solution: Isolte the rdil nd then rise both sides to the nd power. ) ) ) ) By heking both solutions in the originl eqution, we determine tht is not solution. Thereore, the only solution is Emple: Solve the eqution Solution: Isolte the rdil nd then rise both sides to the reiprol power. ) ) 6) ) 6 By heking both solutions in the originl eqution, we determine tht is not solution. Thereore, the only solution is 6

3 Emple: Solve the eqution. Solution: We will rise both sides o this eqution to the rd power beuse the inde o the rdil is. ) Cheking this solution in the originl eqution we my determine tht is solution. Emple: Solve the eqution. Solution: Isolte the rdil nd squre both sides o the eqution. ) ) 6 By heking this solution in the originl eqution we my determine tht is not solution. Consequently, there is no solution. Equtions with Two dils: I n eqution ontins two rdils it my or my not be possible to isolte one o the rdils. I it is possible, then isolte the rdil with the most omple rdind. Emple: Solve the eqution Solution: In this emple both rdils n be isolted by moving one to the other side o the eqution. ) ) By heking this solution in the originl eqution we my determine tht is solution.

4 Emple: Solve the eqution Solution: Isolte the rdil nd then rise both sides to the nd power. ) ) ) ) ) By heking both solutions in the originl eqution, we determine tht both solutions hek. Emple: Solve the eqution Solution: Isolte the rdil nd then rise both sides to the nd power. ) ) ) ) 6 ) ) 6 By heking both solutions in the originl eqution, we determine tht is not solution. Thereore, the only solution is

5 8.6-Applitions: Emple: The digonl o retngulr bo with squre bse n be lulted using the ormul d h where is the length o the side o the squre bse. I the digonl o the bo is 8 in. nd the side o the bo is 8 in., ind the height o the bo. Solution: Substitute the given vlues into the eqution nd solve or h. d 8 8) h h h 8 h 8 h ) 8 h h 6 h ) h ) h h Beuse negtive nswer does not mke sense in the ontet o this problem, the height o the bo is in.

6 Emple: Aording to Einstein s theory o speil reltivity, time will pss more quikly on Erth thn it will in speship trvelling ner the speed o light 86, miles/seond.) The speil reltivity eqution v gives the ging rte o n stronut,, reltive to the ging rte o riend,, on Erth. In this ormul, v is the stronut s speed nd is the speed o light. I the stronut trvels t % o the speed o light or yers, how mny yers will the stronut s riend on Erth hve ged during this time period? When the stronut rehes the speed o light, how muh will he ge reltive to his riend on Erth? Solution: To nswer the irst question, let v. nd Substitute these vlues into the eqution nd solve or. ) v During the yers the stronut hs been in the speship t % o the speed o light, his riend on Erth hs ged pproimtely. yers. When the stronut rehes the speed o light, substitute v into the eqution nd solve or. ) v Consequently, Einstein s theory o speil reltivity seems to suggest mthemtilly the possibility o gelessness or eternl eistene.