MA 103 Sections 7.3, 7.4, 7.5 Simplifying Radical Expressions 1. Radical expressions can be written in simplified form by making use ofthe properties If ~ and ~ are defined, then ~~ = ~ and ~ = nr; n~ For nonnegative a, ~ an ~b Vb m \r:-;; nnr;;- VVI. =a ) (J.VI:: (tv (J W\ 01<. V (J.. 1. Simplify. J2..[fi - m :::.'6 J50 - ~~~.- J2 - ~ fis'ss ~ (e) M j~...... ~ ".... {It - ~l ':=. -t- ( (f) J50J2 '.lv~~ :.:=- '0 Jff% r- -.//'''~ ~' \/ldu'.' -..~... 1.~ -,u... 2. Rewrite in simplified form. Assume all variables are nonnegative. -J700 =- ~fi ::- \~.Ji' (g) 7J18 7 J'fJ't :: 1ls;: y a - _. G) ~5a3 ~8a7 :: [faop)''» ==H [to RO ::: 2. a.s' \ITO 0..." (e) J45 ff{'f::3~ N!: xlof- (h) ~20X5 ~vsjjtf~f ~ 2x. l_~j (k) (;2f;4. '3~ y,j; '"z.j J48 J\b f3".:; if-ff (f) N ~o~ -= ~s~ (i) J;P :: ~,J dj (1).' ~9x3y =0 {?)I.~~ -FY X~ ftk': --3~J
II. To rationalize the denominator of a fraction means to rewrite the fraction in an equivalent form so that the denominator is rational, that is, so that it does not contain a radical. Ifthe denominator contains one term, multiply the numerator and denominator ofthe fraction by the radical in the denominator and then simplify the fraction. 3. Rationalize the denominator and simplify: 10 ~ -~ J5 is - torr ~a ~..ff J7{f S -' - f - ~.: K.ir _ V5 rr '.11---:3 2,.[TO 7.-{fO --,I.:) -- -~ \5" 3JlO m- III. Radical expressions can be simplified by addition or subtraction if and only ifthey are like radicals, that is, ifthey have the sameindex and the same radicand. To add or subtract like radicals, add or subtract the co~fficients. Ifthe radicals are not like radicals, then they cannot be combined by addition or subtraction. 4. Simplify each expression. 7J3-2J3 :: 56' 4V6-~ - 3(r" If-n-+)~ ~5~ 3.JiO +../45 "3 ft+j? +- ffff =- ~ fs +78 =r CJ JS m+5j18 [[fl -{- ~[ffl =~ +(s-~) S?:" ~ ff] Answers l 8 3 10 5 (e) 4/9 (D 5/4 2 1OJ7 3J5 4J3 21J2 (e) x 4 (f) x ' S (g) a 2 2 (h) 2X2~ 3 2J5 6J7 7 (i) x 4 (j) 2a s JlO (k) x 2 (l)3x, J15 3 JlO 15 4 sj3 915 5~ 19J2
MA 103 Sections 8.1,8.2 Solving Quadratic Equations I. Using the Principle of Square Roots; Completing the Square 1. The generalized principle ofsquare roots states that for any real number k, if X 2 = k, then X = ±.Jk,where X is an algebraic expression. Use this principle to solve each ofthe following equations. Answers involving square roots of negative numbers should be written in the form a + hi. 6x~ =13 ~'L 11 --r- x~ =-16 X::t, At f!.; :!:~;o J j( =± J =' 1: It~ :t ~-- (x-3f=17 ~ s,~~'t 1 ~*' k~ ~~~~~ X- '3 =- ± {Ff itvj Ai). ) t 60 t1t s;4 I e,v\ G:3±rrQ 2..Use the method ofcompleting the square and the generalized principle ofsquare roots to solve each ofthe following problems. Solve the equation: x~ +8x-3 = 0 X~i?~ == - -3 X t +1'J. + 1(, ==- 3+ I~ ~+If J- =I~ K+'+ :: ±m,..---'~. _ --!;; -If± ~.. M ri7.f"--) ;= 1M Solve the equation: 2x~ -lox -1 =0 2 X~lot< z(x- L - 5~ L 1x.1.._~y.. + ~.) - (+~ l q.., - 'Z,.. ZlX-~)~ -==. 1- (x-{) = ~ X-?:.. _ -'r-.rt7 1.. - -~7f
Find the x-intercepts of the function f(x) = x~. -4x-7 x~&1f) ± J0<f)~'16)(-ij 'L _;= -- - II. Using the Quadratic Formula Any quadratic equation ax~ +bx+c = 0, a "* 0, can be solved using the quadratic fonnula -b±.jb~ -4ac x=---- 2a 3. Use this fonnula to solve each of the following problems. Write answers in exact form (using radicals). For any answer involving real numbers, use your calculator to approximate the answer correct to three decimal places. Any answer involving square roots of negative numbers should be written in the form a +bi. Solve the equation: 2x~ -3x-7 =0 Solve the equation: x~ -5x+ 7 =0 X~ 3± J(-1) 1-,.-LfLz.)(--j] X:=-(-S)±. {(-rj~lf-(rj(7) 2.b.) ::: 2- X==- +5 i. [-3... 5±f3.z' I ~3t i'0 "2-?...-. ~ x+2 x-4. Let f(x) =-x- and g(x) =-2-' Fmd all x such that f(x) =g(x).
MA 103 Sections 8.3, 8.8 Some Applications of Quadratic Functions 1. The profit made by a small business on the manufacture and sale ofx items is P(x) =_x 2 +126x-1100. What is the value of P(O)? Write a sentence explaining what this number means in the context ofthis problem p{<j)':= - 0 1--{-t1-l.( ())-It00 := -If0 0 1t...- C-(''J t.::i.t/ hllw... a./.055 -t $"00 Algebraically find the number ofitems that should be manufactured and sold in order to realize the maximum profit. What is the amount ofthe maximum profit? F( ~:= - ~'3 t..h'l~(b'3) - \\0<:1 ;: ~'2.-~ (,1 What is a graphing window on your calculator that displays the vertex and all intercepts for this profit function? Using this window, graph the function. ' [0/ 150] '/... [-?.f.nj j 30o~ (e) Suppose that the company is making a profit of $1600. How many items (to the nearest whole number) are being manufactured and sold? ~OO = -)( 'L +- IU)( - l\o~ W QiA.o.~ +,fyrw."j~ Ie -= I'L ~ ± "rru-\'l,~{""'c_-f(~i~)0:-1.7-()~j X1_ t2..' )( -\- \\I:)~ of- \6\J~ - Q "2... )("\.~ \'1..bi 4- ~; C)~ :;. Q '/=- 9Z. 62 ~ 91 r (t) ~ A~' 17 '38 ~'2"" ~. When profit is zero, the company is said to break even. Determine the number of items (to the nearest whole number) that must be manufactured and sold so that the company breaks even. o~ -X"L.+lL6]C-Ho\1 Xl_llb X' +\\OJ.=;:0 'i:;: IL' d: ret') ~ If(I)(rI~~ :=r IICS-C ~ II'? ~ ~ ~ 9 flf ~ 1 :J~
2. If an object is tossed downward with an initial speed (velocity) of v o ' then it will travel a distance of s meters, where s =4. 9t~ + vot and t is measured in seconds. Suppose an object is tossed downward with an initial speed of40 m1sec. (i) How far will it travel in 2 seconds? ~ ~ 1/-.1t 1..+ If 0 t ~(2) := 4.~ (2.-J 1. -4-1foL2.) == ~~., ~ (ii) After how many seconds will the object have traveled 200 meters?.hj- I.{; 9t \.'to t = 200 4-.?t 1..+,+-ot _ '2.,0::) :=; 0 ll$ft a!1j.a~r~~ f«~/)- :D t'\<lyjl ~ tjfl!./ry'r. -t -== - /-0 ± Vto1..-ttCIf.,J ~ 3.5" Sru:..~yo.~. ~(~.q) - ---- Suppose an object falls from a helicopter that has an altitude of 400 m. How long will it take for the object to hit the ground'! ~ f\k+e : ~ =- 0 3. The function h(t) =-16t 2 +80t+ 5 gives the vertical position, in feet above ground level, ofa baseball t seconds after it has been hit. Algebraically determine how high above the ground the ball will rise. t ~ -b ~O. -zo.- 2.(-1") ('1V'j ~?..J ~y t.;: 2.5~~fk h(2-. S') -=--ll(z..sjl.+-8'0(.'2..d+ 5"' =:;;=l05 f~j Ifno one catches the ball, after how many seconds will it hit the ground? Compute an exact answer to this question symbolically and then check your result graphically. Hint: Y What is the height ofthe object above the ground when the object is on the ground? -I ~t. 7..+ %0 t + 5' :; 0 t '::::'-~ ~J~1.._~c... "l1-- Q.::: _ l( 10 -= %0 e:..- =- S /' I. ;- -'60± J%,,1--Z/-GI<.)0') _ /~>~J~~S:.t:... Z(~lb)
=l River 4. A rancher has 3000 feet of fencing available to enclose I a rectangular field that borders on a river. There will be x x ~encing only on 3 sides ~fthe field, as indicated in.the sketch.. fhe area of a rectangle IS A = (length).(width) Wnte a '-----30-0-0--2-x-----J function A(x)that represents the area ofthe field ;4;=- X( '3OQO ~1-)() ~ :-~2-)(-1..-+-'3':ta-oy...--'~ -_._---- Estimate the dimensions of a field that has an area of 500,000 /P. Round answers to the nearest foot. 5"oooo~ == -2.-)('L+-1\lQoo..l goo'g-'u'...-1. &I) 'L. r-:= :1'1. -3ooo~+SQ:) 0'00 =:- 0 _" X== '3= 1: J ( 30'~f '1-('1.)(5"_0"").:;: ~1 (.u.:f]. 3ou~ -''-\J')o I ~-1~2..f~J What are the dimensions ofthe field that encloses the largest area?- -- fn' -Cl. L ll>~\l.~j.fo... -K... tre.d-e..-,.. t A =-_'2.)(.tL+ ';~O~ ------ V' -b -3o'O~ -'3~. n~ <! d~._.u-,,::= z;: = ):=.::: 750 "''.. u.r~ 1- A.SIA'j \""" = 3~ - L "I-- 2.(-'L -If.-=-~ What is the area ofthe largestn~ :::::: 'j1j«) -7...(1 S"o)...-- ::::;~\s_oo~ A 1- '3 ~ -."",= II ----.. f'\~:: -:~1 ~o c:: ~ =} A-== -2i7"';Y~;l_(75"~:::-~;ooo 1& 5. The number ofinmates in custody in US prisons and jails can be modeled by the quadratic function p(x) =_xl +93x+ 1128, where p(x) is the number of inmates in thousands, and x is the number ofyears after 1990. According to this model, in what year will the number of prison and jail inmates in custody in the United States be at its maximum? What is that maximum number ofinmates? FeH.)j =- _!f-cr'1-+91(tf.(. g HI2B :: '3 '2-90.2.q- -Ku'l!)aJ! /.. (f1~ - ~t'102.5o,';..j&1 ] er3>
6. Methane is a gas produced by landfills, natural gas systems, and coal mining operations that contributes to the greenhouse effect and global warming. Based on data from the US EPA, projected methane emissions in this country can be modeled by the quadratic function f(x) =-0.072x 2 + 1.93x+173.9, where f(x) is the amount of methane produced in million metric tons, and x is the number ofyears after 2000. According to this model, what will US methane emissions be in 20091 X~ 200Cf- ~ =7.f( 1) =. - Q,oT1.-C9) 1..-+,.,'] ('1lH71:' ~ ~ S if-j1],4(ji;lm ~c.. ~ In what year will methane emissions in the US be at their maximum? Round to the nearest year. \3 q.o What will be that maximum amount of methane emissions? Pl~ ~ f ==-13, tf ~ ~ Rx)-=: -Q..o7L.x:"L-+!~3)t + [7:-7. -t-(13. Lt) :=: \'6b. <6 f'/fj.b.~y\-lvtjxil---c.; ---1~--'~ What is a graphing window on your calculator that displays the vertex and all intercepts. / r for this function?, \ XM/t\.;::--'?O p\v;;~ '/,:= -o.. o'~t1..+f.~:jx +111,1. x~~ == (2.0 ~ I)JL ~\JM;f"..l.+' ~ ~J'IJJrT fnrrjoqw... y: rei.=: ( -' Y'M~'" =- ~~ ; yf'l\la'1--. ==-ZOJ G~) IZo] '/.. [-5 u.j ~J Answers.1. -1l00 The company will have a loss of $1l00 ifno items are manufactured and sold. 63 items $2869 [0,150]x[-2000,3000] (e) 27 or 99 items (~<f, 1\'7 2. (i) 99.6 meters (ii) 3.5 seconds about 9 seconds 3. Find the vertex of the parabola: (= 2.5 sec., height =h(2.5) = 105 ft fi ul fi d th.... 80+~6720 Use the qua drabc orm a to n e positive (-axis mtercept: t = ~ 5.06 sec 32 4. A(x) =x(3000-2x) x= 191 ft., width = 2618 ft; orx=1309ft., width = 382 ft x =750 ft., width = 1500 ft 1125000 square feet 5. 2036 3,290,250 6. 185.44 million metric tons 2013 186.82millionmetrictons [-50,120]x[-50,200]