Section 4.2: Completing the Square

Transcription

1 Setion 4.: Completing the Square Setion 4.: Completing the Square Objetive: Solve quadrati equations by ompleting the square. In this setion we ontinue to address the question of how to solve any quadrati equation a b 0. Now we will learn a method known as ompleting the square. When ompleting the square we will hange the quadrati into a perfet square that an then be solved by applying the square root property. The net eample reviews the square root property. Eample 1. Solve the equation. ( 5) 18 Use square root property ( 5) 18 Simplify radials Subtrat 5 from both sides Our Solutions COMPLETING THE SQUARE To omplete the square or make our problem into the form of the previous eample we will be searhing for the third term of a trinomial. If a quadrati is of the form b 1 and a perfet square the third term an be easily found by the formula ( b). This is shown in the following eamples where we find the number that ompletes the square and then fator that perfet square trinomial. Eample. Find the value of that makes this epression a perfet square trinomial. Then fator that perfet square trinomial. 8 1 b and our b 8 The third term that ompletes the square is 16: 1 8 (4) Our epression is a perfet square; fator ( 4)( 4) ( 4) Our Answer Page 03

2 Setion 4.: Completing the Square Eample 3. Find the value of that makes this epression a perfet square trinomial. Then fator that perfet square trinomial. 7 1 b and our b 7 The third term that ompletes the square is 49 4 : Our epression is a perfet square; fator Our Answer Eample 4. Find the value of that makes this epression a perfet square trinomial. Then fator that perfet square trinomial b and our 5 b 3 The third term that ompletes the square is 5 36 : Our epression is a perfet square; fator Our Answer SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE The proess in the previous eamples ombined with the square root property is used to solve quadrati equations by ompleting the square. The following si steps desribe the proess used to solve a quadrati equation by ompleting the square along with a pratie eample to demonstrate eah step. Page 04

3 Setion 4.: Completing the Square Steps for Solving by Completing the Square 3 Eample Separate the onstant term from the variable terms If a 1 then divide eah term by a. 3. Find the value that ompletes the square: 1 b. 4. Add the resulting value to both sides of the equation. 5. Fator the perfet square trinomial. 6. Solve by applying the square root property ( 3) 11 ( 3) The advantage of this method is that it an be used to solve any quadrati equation. The following eamples show how ompleting the square an give us rational solutions irrational solutions and even omple solutions. Eample 5. Solve the equation by ompleting the square Separate the onstant term from variable terms Subtrat 48 from both sides of the equation 0 48 Divide eah term by 10 4 Find the value that ompletes the square: ( 5) 1 Our b 10 ; 1 10 (5) 5 Add 5 to both sides of the equation Fator the perfet square trinomial Solve by applying the square root property 1 b Page 05

4 Setion 4.: Completing the Square ( 5) 1 Simplify radials 5 1 One equation for one equation for 51 or Subtrat 5 from both sides 4 or 6 Our Solutions Eample 6. Solve the equation by ompleting the square Separate the onstant term from variable terms Add to both sides No need to divide sine a 1 Find the value that ompletes the square: Add 9 4 Our b 3 ; to both sides of the equation 1 b 9 9 Need ommon denominator (4) on right side of equation Fator the perfet square trinomial Solve by applying the square root property Simplify radials Add 3 to both sides of the equation 3 3 Notie that we already have a ommon denominator 3 17 Our Solutions As Eample 6 has shown when solving by ompleting the square we will often need to use frations and be omfortable finding ommon denominators and adding frations together. Sometimes when solving quadrati equations the solutions are omple numbers as is the ase in Eample 7. Page 06

5 Setion 4.: Completing the Square Eample 7. Solve the equation by ompleting the square Separate the onstant term from variable terms Subtrat 30 from both sides of the equation No need to divide sine a 1 1 Find the value that ompletes the square: b Our b 6 ; 1 6 ( 3) Add 9 to both sides of the equation 6 91 Fator the perfet square trinomial ( 3) 1 Solve by applying the square root property ( 3) 1 Simplify radials 3 i i 1 Add 3 to both sides Our Solutions We an use ompleting the square to solve any quadrati equation so we want to get omfortable using the si steps of this method. Page 07

6 Setion 4.: Completing the Square Pratie Eerises Setion 4.: Completing the Square Find the value of that makes eah epression a perfet square trinomial; then fator that perfet square trinomial. 1) 30 5) 15 ) a 4a 6) r r 1 9 3) m 36m 7) y y 4) 34 8) p 17 p Solve eah equation by ompleting the square. 9) ) ) n 8n ) v 8v ) b b ) ) 3k k ) 4z z ) 8a 16a ) ) p 16 p ) n 16n ) m 8m 1 0 1) ) 6r 1r 4 6 3) 8b 16b ) 6n 1n ) v 14v 36 7) 3k 9 6k 8) 5n 9) 30) 10n 15 p 8p ) 7n n 7 7n 6n 3) n 33) 8n 4n 1 16n 64 34) b 7b ) ) m 15 9m 37) 4b 15b 56 3b 38) 10v 15v 7 4v 6v 39) n 1 10n 40) a 56 10a Page 08

7 Setion 4.: Completing the Square ANSWERS to Pratie Eerises Setion 4.: Completing the Square 1) ) 3) 4) 5; ( 15) 144; ( a 1) 34; ( m 18) 89; ( 17) ) 4 ; 1 1 6) ; r ) ; y ) ; p 4 9) ) ) 4 i 9 4 i 9 1) 1i 4 1i 4 13) 14) ) ) ) ) ) 9 7 0) ) 1 i 1 1 i 1 ) 1 3 3) 3 7 4) 3 1 5) 5 i 5i 6) ) 7 3 8) ) 1i 1 i 30) ) 4 i 39 4 i 39 3) ) 71 34) 6 35) 4 36) ) ) ) ) Page 09

8 Setion 4.: Completing the Square Page 10