FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.

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1 Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is defied to be the oegative umber a where a 2 = x. Higher Order Idices idex radicad x = a if a = x Note: If is eve, we require that x 0. Ex 1 Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) ) o) p) q) r) s) t) u) v) w) x) y) z) STUDY cube roots ad fourth roots o page 571. Page 1 of 21

2 Ex 2 Evaluate 9. Next, solve for x: x 2 = 9 usig two methods. Note the differece. Square root of a umber (whe symbol is used) is always + or 0. The ( ) sig is itroduced whe solvig equatios. eve Note must be at least 0 Aswer must be 0 eve egative Aswer is ot real odd aythig Aswer CAN be egative Ex Simplify. Assume that all variables represet oegative umbers. a) x 4 b) x 0 c) 64x 6 d) 64a 15 b 21 4 e) 16x 8 Ex 4 Sec 8.2: Simplifyig Radicals Substitute the give umbers ito the expressio b 2 4ac ad the simplify. a) (#2) b) (#6) a = 6, b = 7, c = 5 a = 7 4, b =, c = 2 4 Rules ab = a b a = a, b 0 b b If is eve, we require that either a or b is oegative. Page 2 of 21

3 Ex 5 Simplify. Assume that all variables represet oegative umbers. a) x b) 27 c) 60 d) 52 e) 16x 4 y 6 f) 12y 5 g) 25x h) 25x i) 8x 5 y 2 5 j) 64x 10 k) x 2 + 4x + 4 l) 6a 4 b 6 c m) 121a 8 b 5 c 1 ) 294x 2 4 o) 2x 9 y 10 5 p) 64x 8 y 4 z 11 q) 16 Page of 21

4 Ex 6 Simplify. Ratioalize the deomiator if ecessary. a) b) c) 5 w 8 x d) e) f) g) 1 11 x 2 x 2b 2 ab 5 5 h) i) j) k) y xy 2 50x2 y z Ex 7 Simplify. Do ot assume the variables represet positive real umbers. a) b) c) 49x 2 40x y 2 16x x + 25 Page 4 of 21

5 Sec 8.: Radical Expressios *Thik LIKE RADICALS (similar to LIKE TERMS)* Ex 8 Combie if possible. a) b) PP 7 xy 5 y 9 xy 12 y c) d) e) 6 5y 20y f) x x 2 g) 2x 45 7x 80 h) 2x 50x 18x i) j) k) l) x + 1 x Page 5 of 21

6 Sec 8.4: Multiplicatio ad Divisio of Radicals Ex 9 Simplify ad leave aswers i expoetial form. Assume all variables represet oegative real umbers x 2 7x 2 (7x) 2 ( 8) 2 Rule For ay oegative umber a, a a = a 2 = a. NOTE that a 0 Recall a b = ab a = a, b 0 b b (Note that a ad b are NOT both egative if is eve.) Ex 10 Cosider the followig: x 2 + y 2 = x + y Extra Credit: Prove or disprove Ex 11 Multiply ad simplify all aswers. (Do ot use calculator or table of square roots.) a) 2 b) c) 2 7 d) 8b 12 Page 6 of 21

7 e) ( 2 a 5a) ( 20 9 a 56a) f) 2(5 w 2 ) g) (5 6 2)( 6 + 2) h) ( 6 + 5) 2 i) (5a 2)(5a + 2) j) 17x 4x k) 7( ) l) 12( ) m) ( x + 2) 2 ) ( x + + 5) ( x + 5) Page 7 of 21

8 Ex 12 Simplify ad ratioalize deomiators whe ecessary. a) b) c) x cojugate Expressio Cojugate Examples x + 2 2x x 9 2x + 5 2x + 5 d) e) DO x 6 6 x 6 6 Page 8 of 21

9 *Start by ISOLATING THE RADICAL!!* AND Sec 8.5: Radical Equatios ad Graphs Ex 1 Solve for the variable. We MUST check for extraeous solutios. a) x + 2 = 6 b) 5x 5 4x + 1 = 0 c) 12x + 7 = 2x d) x + 4 x = 4 e) 2x + 2 = x f) 2x 2x + = 9 Page 9 of 21

10 g) y + 16 = 2y + 2 h) x = 4 i) x 2 + = 4x + 1 j) 2x 1 = x Page 10 of 21

11 Recall Pythagorea Theorem I ay right triagle, if c is the legth of the hypoteuse ad a ad b are the legths of the two legs, the a 2 + b 2 = c 2. b c a Ex 14 The hypoteuse of a right triagle is 25 meters i legth. Oe leg is 5 meters shorter tha the other leg. Fid the legth of each leg. Ex 15 Simplify ad ratioalize deomiators whe ecessary. a) b) c) PP d) x 22 2x x 2x x 5 Review each approach. Complete Additioal Practice with Ratioalizig (Expressios) ad Solvig Radical Equatios hadout for test!! Posted o website. Page 11 of 21

12 Sec 9.1: Factorig ad Square Root Property AND Sec 9.: The Quadratic Formula Def A quadratic equatio is a polyomial equatio of degree two. The stadard form of a quadratic equatio is ax 2 + bx + c = 0, where a 0, ad a, b, ad c are real umbers. Ex 16 Write i stadard form. Determie the values of a, b, ad c. x x + 8 = 17x 2 Ch 9 4 Methods of Solvig ax 2 + bx + c = 0 Zero Factor Property (ZFP) Try this first if the quadratic equatio is i stadard form: ax 2 + bx + c = 0 ex 6x x 7 = 0 If ZFP does t work: Quadratic Formula (QF) This should be your LAST resort. Try this whe method 1 fails (whe the quadratic expressio does t factor). ex x 2 + x 1 = 0 OR Square Root Property ( Prop) Use this whe the quadratic equatio is missig the x term or whe it s i the form ( ) 2 = some # ex x 2 75 = 0, (x 7) 2 = 15 There are 4 methods of solvig a quadratic equatio. They are listed i the order i which you should try each method. *Completig the Square (CTS) Use this method ONLY whe required to. *math120 Ex 17 Solve for x. 6x 2 = 24 Rule: x 2 = Square Root Property ( Prop) If x 2 = a, the x = ± a for all a 0. Note: x = ± a meas x = a or x = a. Page 12 of 21

13 Sec 9.-This sectio itroduces a method for solvig quadratics ad ca be derived from completig the square. Quadratic Formula The roots of ay quadratic equatio of the form ax 2 + bx + c = 0, where a, b, ad c are real umbers ad a 0, are x = b± b 2 4ac. 2a Note: Useless uless memorized perfectly. Ex 18 2 ± 6 6 Simplify completely, if possible. If it caot be simplified, rewrite the problem i a differet form. 12 ± ± 50 5 ± 10 2 ± ± ± PP 2 ± Bob 2 ± ± 4 4 ± 25 4 Ex 19 Solve 2x x + 45 = 0 usig the QF ad ZFP. QF ZFP 2x x + 45 = 0 2x x + 45 = 0 Page 1 of 21

14 Ex 20 Solve usig two differet methods. ( 2 2 x 7) = 16 ( 2 2 x 7) = 16 QF (discuss/brief) Prop Ex 21 Sec 9.: The Quadratic Formula Choose the most efficiet method to solve for the roots of each quadratic equatio. (Solvig for If there are o real roots, say so. roots meas settig equatio equal to 0 the solvig for the variable.) a) 6x = 18x 2 b) x 2 = 72 Do Do c) x 2 2x = 4 d) x 2 + x 40 = 0 x = 5, 8 Do Prac Prob Page 14 of 21

15 e) (x 6)(x + 1) = 8 x = 7, 2 f) 4x 2 12x + 9 = 12 Prac Prob Do g) x 2 4x + 2 = 0 h) 4(x 1) 2 = 5 8x x = 1, Do Do i) x 2 = 25 j) x 2 + 9x + 7 = 0 x = 9± 5 x 2 = 25 Prac Prob x = ±5 2 Page 15 of 21

16 k) (111x 14) 2 = 4 l) 4x 2 12x + 9 = 0 Do (QF?) m) x 2 2 = 4 ) 10x(x + 2) = (x + 2) x = 2, 2 5 Prac Prob o) 2 x2 x = 5 2 Start: Good IC prob p) 2(2x + 1)(x + 1) = x(7x + 1) Do Page 16 of 21

17 q) 2x x = 0 x = 0, 11 Prac Prob 2 r) (6x + 2)2 = 12 x = 2±2 = 1± 6 Prac Prob s) x 2 2 = 0 t) (b + 4)(b 7) = 4 u) v) w) x 1 = 20 2x + 5 = 2 5x + 7 x 2 x 1 x 2 9 = 1 + 2x 8 x Do Prac Prob x = 8, 5 Page 17 of 21

18 x) y) z) 2x 2 4x 5 = 0 (a 15) 2 = 49 9x 2 6x + 1 = 2 As time permits/pp As: x = 2± 14 2 a') b ) c ) 100x 2 600x = 0 (x 4) 2 + (x 2)(x + 4) = 12 1 x x + 4 = 1 Page 18 of 21

19 d') e ) f ) 8t 5 + 2t 4 = 12t 0.0x 2 0.0x 0.0 = x = 2 x 2 Ex 22 For the followig quadratic equatios, choose the method that works best. Do NOT solve. Provide a systematic approach to each problem. a) x 2 x + 6 = 0 b) x 2 4x = 2 c) x 2 = 4 d) (x + 1) 2 = 12 e) (x + 1) 2 x = 12 f) x 2 = 6x Ex 2 How ca we tell if a polyomial is prime or if a quadratic equatio ca be solved usig the ZFP? Page 19 of 21

20 Ex 24 A perso stadig close to the edge o top of a 80-foot buildig throws a baseball vertically upward. The quadratic equatio s = 16t t + 80 models the ball's height above the groud, s, i feet, t secods after it is throw. How may secods does it take util the ball fially hits the groud? Ex 25 (PRACTICE PROBLEMS FOR EXAM complete day of review) Solve each equatio. a) 1x x = 0 b) x x 2 = 4 c) 9x 2 + 5x 10 = x d) 7 x x = 1 Page 20 of 21

21 Ex 26 Solve by completig the square. Outlie steps. a) x 2 12x 1 = 0 b) 2x 2 4x 5 = 0 Steps to CTS c) x 2 9x = 0 d) 2x 2 + 1x = 7 Page 21 of 21