10.2 Solving Quadratic Equations by Completing the Square

Transcription

1 . Solving Qadratic Eqations b Completing the Sqare Consider the eqation ( ) We can see clearl that the soltions are However, What if the eqation was given to s in standard form, that is 6 How wold we go abot solving the eqation? What we want to do is change the eqation from standard form to the etracting roots form. That wa we can easil solve. To do this we se something called completing the sqare. Completing the Sqare To complete the sqare for the epression b b ( ) we add b ( b) ( b). We get Notice that when we add this completing the sqare piece, we get a perfect sqare trinomial. This will alwas be the case, therefore, we will alwas be able to factor the epression once we have added in the epression ( b). Eample : Complete the sqare on Soltion: We simpl add on the ( ) Factoring we get. Then factor accordingl. piece. In this case b b. So we get ( ) ( ) Notice that when we complete the sqare, the reslting trinomial will alwas factor the same wa. That is, ( variable /- inside ( ) of completed sqare piece ). We know if its or b taking the same sign as the original b vale. So eample we can visalize like the following variable ( ) ( ) sign inside the parenthesis Again, when we complete the sqare it will alwas factor like this. So we merel need to remember this pattern for more complicated problems.

2 So we reall want to se completing the sqare to solve qadratic eqations. To do that we first will notice a few things. To complete the sqare, the leading coefficient mst alwas be as it is above (if its not we can easil make it ) and completing the sqare will work for an qadratic eqation. This means that we can alwas se completing the sqare as a techniqe for solving. Solving a b c b Completing the Sqare. Isolate the variable terms and make the leading coefficient b dividing each term b a.. To preserve the eqalit, add the epression ( b) to both sides of the eqation.. Factor the variable side of the eqation. It will alwas factor as a perfect sqare.. Solve b etracting roots. Eample : Solve the eqations b completing the sqare. a. 6 7 b. c. Soltion: a. To solve b completing the sqare we follow the steps above. We get ( 6) 7 ( 6) ( ) 7 ( ) So the soltion set is {, } b. Again, we proceed as follows. Isolate the variables Add ( b) to both sides Factor as shown above Finish b etracting ( ) ( ) ( ) ( ) So the soltion set is {, }. roots Note: The previos qestion shows s the advantage to sing the formla ( variable /- inside ( ) of completed sqare piece ) to factor the trinomial portion. It simplifies the factoring which wold have been mch more difficlt in this case. So we can instead jst

3 remember the pattern and se it to factor all completing the sqare problems no matter how difficlt or trivial. c. Finall, we will again complete the sqare to solve. However, this time we need to make sre the leading coefficient is. So, we will start b dividing each term b and then proceed as sal. ( ) ( ) ( ) ( ) 6 In the last step, the radical was simplified b rationalizing the denominator. So or soltion set is { },. Now that we have the basic idea of completing the sqare, lets see some eamples which are a little more challenging. Eample : Solve the eqations b completing the sqare. a. b. ( )( ) Soltion: a. The first thing we mst do is write the eqation in standard form. So we move the terms arond to get So in this eqation its reall eas to jst start completing the sqare withot remembering that we need the leading coefficient to be. It is crrentl. So we can fi that b mltipling the entire eqation b. We then proceed as normal. ( ) ( )

4 ( ) ( ) So or soltion set is { },. b. On this eqation we need to again start b getting rid of the parenthesis. Once we do that we can contine as normal ( )( ) ( ) ( ) ( 7) ( 7) Notice that we can actall perform the operations of and in the last step. Therefore, we mst. So we have 7 6, So or soltion set is { }. 7 6 The fact that we got rational nmbers in the answer of the last eample tells s that factoring cold have been sed to solve the qadratic in that problem. However, since the instrctions told s to solve b completing the sqare, we had to solve sing completing the sqare anwa.. Eercises Complete the sqare on the following. Then factor accordingl t t t. t Solve the eqations b completing the sqare t t... t t.. 6

5 t t. t t. r r z z 7.. v 6. t t ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) 7 ( ) ( ) ( ) ( ) ( 6) ( )