Ch3 Section 3 Notes Answers

Completing the Square For use with Exploration 3.3 Essential Question How can you complete the square for a quadratic expression? EXPLORATION: Using Algebra Tiles to Complete the Square Go to BigldeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to complete the square for the expression x 2 + 6x. a. You can model x 2 + 6x using one x 2 -tile and six x-tiles. Arrange the tiles in a square. Your arrangement will be incomplete in one of the corners. b. How many 1-tiles do you need to complete the square? c. Find the value of c so that the expression � 1.J J.J J.J is a perfect square trinomial. d. 0 ( 1 Write the expression in part (c) as the square of a binomial. EXPLORATION: Drawing Conclusions r··-·11 -- -- )( ')t 'X \ ) ) J ;'. I • l Work with a partner. a. Use the method outlined in Exploration 1 to complete the table. Expression x2 + 2x #: I 1• 'C' ' x + 4x .4?.;c-. 2 • r,+f.' •c I x2 + 8x -�I; x2 + lOx + IS> ' ' _,,- Value of c needed to complete the square I Expression written as a binomial squared ()('·t t,,I ( .,._ +- ?..>) 1,, I f.o ( >c r- �1\2. "' 25 b. Describe how you can solve the quadratic equation x t"'\1. { X +-fh 2 1 by completing the square. + 6x = �(3> 1,. \ tCJ ;: I +c, Algebra 2 I\) \ 55 Notetaking with Vocabulary For use after Lesson 3.3 In your own words, write the meaning of each vocabulary term. completing the square Core Concepts Completing the Square Words To complete the square for the expression x2 + bx, add (%J. Diagrams In each diagram, the combined area of the shaded regions is x2 + bx. Adding (%J2 completes the square in the second diagram. xi I I I X x 2 I+I X l--b---1 X X 2 + bx + er (!}x 2 I I I (�)x /(!} : , _____ _! Algebra x2 ----' t"P\p\e� �� $��,_ = (X + �)(X + �) = (X + �)' Algebra 2 .,.,, 57 Extra Practice In Exercises 1-3, solve the equation using square roots. Check your solution(s). 1. x 2 + 4x + 4 = 2 f;_e, tv,r { � /2-)..� �/M, �,,-�..._Jy +- 2 x+-z.. � n -'L -'2- ,_J ) X :. - i ± J 3. l' 9w 2 + 6w + l � 2-C) + zo =. l O J .f ·2.0 / l ::: 20 ±-1orij 1 = -18 2�(}') fq_ C. t-<-r - v(1w + ,\1 :f- S i-v +I:: -I .-:i JI,\,"� 3 I � / � �·-(1t10--I t -1+-3"J:; ..- (, -- - 3 ;, j-t±3ifi-. , --�-_J -,-- In Exercises 4-6, find the value of c that makes the expression a perfect square trlnomial. Then write the expression as the square of a binomial. 4. y2 - 14y + s. s 2 + 17s + c C l q;'" /�2. 6. 58 \_ z2 Algebra 2 - .., I r_ In Exercises 7-12, solve the equation by completing the square. 7. r2 - 6r - r 1. �1o,., '\, t.A>t'�') 'l. -'=, r -1- �-;. --z.� ----?(_ + /0'?( '2. -=- z_ r s. x2 + lOx + 28 = 0 = 0 .J.. 2. +2 2 �Ls�,._ x "L- +10-y: +2s- � -z� +:is­ Fca..w �-JII �t.� f3 -I + 1) = � - Y +r - �, i.'-,)li)" iq y -1- fi =--� F--'1.cfur p:i; (:3 J_ I ' 3 i� � -5 - s- - - - )�,. -s ± ih] 10. 3 ::: -; )( +s :: t. v.-- J D,s�:hMk '2. -i- J-( >< t 5') ,,-,; ��JTT 9 . y (y -=- -2..-y 'L 2 +'1 r-�::!Ti � -Z-� -z. 2t 2 + l 6t - 6 = 0 z_. l 2- t; '2.. + &°"t: - � �· c) t ) f-) ii.+bt. -1-� • � s ( y �['t)\ --;.-3 4t,I)� t �..J-�t -Fc:t +I(;:,� S +.,I� �·r"'- X +t.t ; � l -Lr - l.\._ X ;_ -L[ Algebra 2 tM I 59 In Exercises 7-12, solve the equation by completing the square. 3x(2x + 10) = -24 Distri b"' k 11. -r; x� + -" • ,,I , X. : " I 12. I 4x2 1,.1 Sx + 28 = 3x 2 + x -°3*� - X X1... -I - - l i /X / b � + i., x,., -�x -z._� � l·'3\ii. X :. 0 -2.-� � - 2� i x - , )( ..fei ; -2-i +-'1 X -� :: i,J1 -f � t�S �!)tw, 2 13. Explain how the expression (4p + l) + 8(4 p Then write the expression as a square of a binomial. I� wt- I;\ �e.. x.. = &.fp. ,� l'I'\ t r. lt-L + l) + 16 is a perfect square trinomial. e >< p-re r o�, s 1 x 1 ,- � -x. , I � \lv�ttA p� # t ( t" .. , , So fl,..e_ -A· A"'- l e � r') re� �i'()n [0p+-1) t 60 Algebra 2 I,.,] ... A·l s t) fJtl'fft'+- i ·� s1(A�. r-e. f''--t t � f(-;t\DM,'4) ?' Ct:,.. (lr-f., In Exercises 14-17, determine whether you would us (actoring, square roots completing ,) ,, the square to solve the equation. Explain your reasoning. The�olve the equation. -14. x 2 + 7x = 0 15. Fctct-ot K('K+-7\-:;0 (x - 1) 2 5q t,t'1 v-l. r: = 35 p::/3___. f'.--oo +- 1 S- '- 0 ..,_-I� J +t => -f I (X 17. 4x 2 + 8x + 12 Fttd--w' 2'i (x +-2'l<- + s1 :.- 0 x 2 +Z;ef-1:> 18. = 0 -=- cJ J The area of the triangle is 30. Find the value of x. 1 bh A.,,;i:.--51, '; ; i (x+,.>)(3 ') 3x fu d.,., x+I :-ifi '+:: �h.:c g� c�}· >T g � 0 :. :; >< t. O :. >x 1 . G::- () � o: +� )<.' -3 x +'- 'le •le() "3> i.. - •• (r ("l � l Wi pit k '"Y (O'0,-l.. - I ,y:-_ ��ii� t-0 � X 1.. + z.4 -1±,:J'2--· 1 L\ )t.· 3 +z.. >< -'20,,,...,. �,.I, 1. � 't , I () '-1,5" Nv� F�c..\of �ble 61 19. Write the quadratic function f (x) x + 6x + 22 _f (?C')-Z2- �· .t (?Cl ·- 2 2 -- l{_�, � 2 in vertex form. Then identify the Yertex. t"0,,.. l / � e 'i 5 x--i. + lo f z._4(�\l· + 6 y +· 't Fetc.+o, 4 "\ ::: X f (ix'\ - t + r� ') � L ( >< +- �5")7+ -- 1 ,) _ ... gv \)ede.'f- ,�(-�>I� 20. A golfer hits a golf ball on the fairway with an initial velocity of 80 feet per second. The height h (in feet) of the golf ball t seconds after it is hit can be modeled by the function h(t) = -16t2 + 80t + 0.1. a. Find the marimum height of the golf ball. f v-c,-. / ) F' Ad I a i- -,.. C. ·I l--e.. A 0 _; '" h (?. s\ ";, .. u,(2. s\ + �,o{ 2, ·s")+, I � -lb (�.2 SJ+. 1.c>o -I, I � -100 �200+,I b. How long does the ball take to hit the g(o{nd? _,_ ��,��t· 0 .f. /QO -/-, I 0 0 -:.- - IC? .f '2. • t- &-0-t + I N° I vv' vV"l t Po( (\gf tJ 0t F;,� c +tH "I 6 1-( / :;';'1 �; :, ,'t:... tc ii+ Algebra 2 o+ I t'� I/,/ ( 0,..,, pl I le. ·tLe_ s-. -'; i't t-t< i :i rf'�II .-1311; I 62 /1\