10.4 Solving Equations in Quadratic Form, Equations Reducible to Quadratics

Transcription

1 . Solving Eqations in Qadratic Form, Eqations Redcible to Qadratics Now that we can solve all qadratic eqations we want to solve eqations that are not eactl qadratic bt can either be made to look qadratic or generate qadratic eqations. We start with the former. Eqations of Qadratic Form An eqation of the form a b c eqation is qadratic form. where is an algebraic epression is called an An eample of an eqation in qadratic form wold be. The wa to visalize this is b sing the properties of eponents. We cold see it as So the epression ( ) ( ) is the algebraic epression referred to in the above bo. Another eample of an eqation is. We can visalize it this wa This time the epression is. ( ) We generall find the epression involved b looking for something that shows p to the first power and then also to the second. We solve these eqations b sing a simple sbstittion procedre. We will eplain this b wa of eample. Eample : Find all soltions to the following eqations. a. b. Soltion: a. B looking at the work we did above we know ( ) ( ) So we make a sbstittion as follows: Let. If we do this, then b sbstitting this into the eqation on the right we get This is clearl an eqation that we can solve. It s jst qadratic. Therefore we will solve. Lets do so b factoring. We get ( )( ) However, the original eqation was in terms of. Therefore we mst solve for. So we recall or sbstittion. Re-sbstitting we have

2 Now we simpl solve b etracting roots. So or answer is ± ±,,,. So or soltion set is { } b. Again b recalling the work done above we have ( ) So again, or sbstittion is. Sbbing this into the eqation on the right we have We will solve b factoring Re sbbing ( )( ) and solving like before we get However, recall that when sqaring both sides of an eqation, we mst check or answer for etraneos roots. So we go to the original eqation and check them there. Check : Check : () ( ) checks does not check Since the does not check it is etraneos and ths thrown ot. So the soltion set is { }. So b the eample we get the following method Solving an Eqation in Qadratic Form. Determine the sbstittion to be sed. Make the sbstittion and solve the reslting eqation.. Re-sbstitte the epression from step and solve for the original variable.. Check or answer if necessar. Lets see some more eamples. Eample : Find all soltions to the following eqations. a. z z b. c. ( ) ( )

3 Soltion: a. We need to first determine the sbstittion. We are looking for an epression in the eqation that is both sqared and to the first power. Notice that b properties of eponents. So therefore we se the sbstittion z z z z z z z. So we have So we now solve the qadratic, re-sb and solve for z as follows ( )( ) Re sb z z z Cbing both sides z z Checking or answers reveals that the are both good answers. So or soltion set is {, }.. b. Again, we start b determining the sbstittion we shold se. Notice ( ) So let and solve as above. We get ( )( ) So we need to solve the bottom eqations. To do this we will get rid of the negative eponents and then solve b clearing fractions. So or soltion set is {, }. c. This time it is ver clear what or sbstittion shold be. Clearl let. Sbbing and solving like before and solving we have

4 ( ) ( ) ( )( ) ( )( ) ( )( ) So or soltion set is { }.,,, The other tpe of eqation we wanted to solve was eqations that generate qadratic eqations. This sall happens on radical or rational eqations. Since we have discssed solving these tpes previosl, we will merel refresh or memories on the techniqes sed. Eample : Find all soltions to the following eqations. a. b. Soltion: a. To solve this eqation we recall that to solve a radical eqation we mst isolate a radical and sqare both sides as man times as needed ntil all the radicals are removed (c.f. Section.). So we do that process here. ( ) ( ) ( ) ( ) ( ) ± Again, since we raised both sides to an even power, we mst check or answers.

5 Check : Check : ( ) ( ) ( ) ( ) Both soltions check. So or soltion set is {, }. b. To solve this eqation we recall that to solve a rational eqation we mst mltipl both sides b the least common denominator then solve the remaining eqation (c.f. Section.). In this case the LCD is ( )( ). So we solve as follows ( )( ) ()( )( ) ( ) ( ) ( )( ) ( ), Since we started with a rational eqation (which has a limited domain) we mst make sre that we have no etraneos soltions. To do this we simpl need to check that neither soltion makes the origonal eqation ndefined. That is, does either soltion give the origonal eqation a zero in the denominator. Since it is clear that the onl vales that make the denominator zero are and, or soltions are not etraneos. Ths or soltion set is {, }. Notice that the qadratic eqations in or eamples all factored nicel. If the eqation does not factor simpl se the qadratic formla to solve. If the eqation reqired a sbstittion to solve, be sre to re-sbstitte and finish solving after sing the qadratic formla. To check the answers simpl se a decimal approimation.. Eercises Find all soltions to the following eqations z z. a a.. d d..

6 t t... t t..... ( ) ( )..... ( ) ( ). ( ) ( ).... ( ) ( )... ( ) n. n n.

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