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# Recent problems solved by 'Theo'

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 Probability-and-statistics/398939: if there are 90 pizzas, and each person eats 1/6 of the pizzas--How many people are there?1 solutions Answer 282577 by Theo(3464)   on 2011-01-24 14:25:54 (Show Source): You can put this solution on YOUR website!each person eats 1/6 of a pizza. 6 people per pizza 90 pizzas number of persons is equal to 6 * 90 = 540. formula is: number of people * amount of pizza eaten per person = number of pizzas eaten. n = number of people r = amount of pizza eaten per person p = number of pizzas eaten n * r = p r = 1/6 because each person eats 1/6 of a pizza. p = 90 (given) n * r = p becomes: n * (1/6) = 90 multiply both sides of this equation by 6 to get: n = 90 * 6 which becomes: n = 540 540 people eating 1/6 of a pizza each consume 90 pizzas (540 * 1/6 = 90)
 Sequences-and-series/398922: Find S8 for the geometric series 256 + 64 + 16 + 4 +…1 solutions Answer 282575 by Theo(3464)   on 2011-01-24 14:19:29 (Show Source): You can put this solution on YOUR website!this is a geometric series with a common ratio of 1/4. formula for sum of a geometric series is: Sn = (a*(1-r^n)) / (1-r) n = number of terms in the series. S = sum of the terms in the series. a = the first term in the series. r = the common ratio in the series. In this problem: n = 8 a = 256 r = (.25) Substitute in the formula to get: Sn = (a(1-r^n)) / (1-r) becomes: S8 = (256 * (1 - .25^8)) / (1 - .25) Simplify to get: S8 = (256 * (1 - .000015259) / (.75) Simplify further to get: S8 = (256 * .999984741) / (.75) Solve to get: S8 = 341.328125 You can verify this is correct by detailing each individual term from 1 to 8 and then summing them up. You get: S1 = 256 S2 = 64 S3 = 16 S4 = 4 S5 = 1 S6 = .25 S7 = .0625 S8 = .015625 Sum is equal to 341.328125 Answer is the same. You're good.
 Mixture_Word_Problems/388140: Write a linear equation to model this situation. An alloy is a mixture of metals. An artist was commissioned to make a 100g bracelet with a 50% silver alloy. He has a 60% silver alloy and a 35% silver alloy. Solve this using elimination or substitution. Help!1 solutions Answer 274441 by Theo(3464)   on 2010-12-20 19:04:12 (Show Source): You can put this solution on YOUR website!bracelet will be 100 grams with a 50% silver alloy. artist has 60% silver alloy and 35% silver alloy. he needs to combine these together to make the bracelet with 50% silver alloy. x = number of grams of 60% silver alloy. y = number of grams of 35% silver alloy. x + y = 100 grams of bracelet .60 * x = amount of silver in the first alloy. .35 * y = amount of silver in the second alloy. .60 * x + .35 * y = .50 * 100 grams of silver in the bracelet your 2 equations that you need to solve simultaneously are: x + y = 100 .60*x + .35*y = .50*100 we'll use substitution. from the first equation, solve for y to get y = 100 - x substitute 100-x for y in the second equation to get: .60*x + .35*(100-x) = .5*100 simplify to get: .60*x + .35*100 - .35*x = .5*100 simplify to get: .60*x + 35 - .35*x = 50 combine like terms to get: .25*x + 35 = 50 subtract 35 from both sides to get: .25*x = 50-35 simplify to get: .25*x = 15 divide both sides by .30 to get: x = 15/.25 = 60 if x = 60, then y = 100 - 60 = 40 you need 60 grams of 60% alloy and 40 grams of 35% alloy. 60 + 40 = 100 grams of alloy in the bracelet (from x + y = 100). .6*60 + .35*40 = 36 + 14 = 50 grams of silver in the bracelet (from .6*x + .25*y = .5*100). Your answer is: He needs 60 grams of 60% alloy and 40 grams of 35% alloy to make the bracelet.
 Permutations/388108: There are 6 boys and 4 girls and a group of 5 people must be formed. How many groups are possible consisting of 2 girls and 3 boys Answer: nCr=combination of n things taken r at a time= n!/((n-r)!*r!) 4C2 * 6C3 =6*20 =120 groups If a group is chosen randomly from the 10 people, find the probability that the group will consist exactly of 2 girls and 3 boys. 1 solutions Answer 274440 by Theo(3464)   on 2010-12-20 18:47:45 (Show Source): You can put this solution on YOUR website!The probability that a group of 5 chosen out of a group of 10 will contain exactly 2 girls and 3 boys is equal to: (C(6,3) * C(4,2)) / C(10,5) That becomes 120 / 252 = .476190476 That is the number of ways you can get a group of 5 consisting of 2 girls and 3 boys divided by the total number of ways you can get a group of 5. It can also be calculated another way. The probability of getting bbbgg (3 boys picked first and then 2 girls) would be: 6/10 * 5/9 * 4/8 * 4/7 * 3/6 This becomes .0476190476 You then need to multiply that by the number of different ways the group of 5 can be chosen. That would be C(5,2) = 10 The total probability is therefore .0476190476 * 10 = .476190476 The number of ways the group of 5 can be choen is shown below: bbbgg bbgbg bbggb bgbbg bgbgb bggbb gbbbg gbbgb gbgbb ggbbb bbbgg means boy drawn first and second and third girl drawn fourth and fifth. bbgbg means boy drawn first and second girl drawn third boy drawn fourth girl drawn fifth. etc. until you get to: ggbbb means girls drawn first and second boy drawn third fourth and fifth. both ways will get you the same answer.
 Quadratic-relations-and-conic-sections/388126: y=1/3x^2-2x+7 I need to find the vertex. I know I first use the axis of symmetry which is x=-b/2a. I did that and I got 2/2/3. I don't know what to do next. Please help. Thank you.1 solutions Answer 274439 by Theo(3464)   on 2010-12-20 18:24:55 (Show Source): You can put this solution on YOUR website!y = (1/3)*x^2 - 2*x + 7 x = -b/2a ------------------ a = (1/3) b = -2 c = 7 ------------------ -b/2a = -(-2)/(2/3) = 2/(2/3) = 2*3/2 = 6/2 = 3 ------------------- you found x now you need to find f(x) to get the y value when x = 3, (1/3)*x^2 - 2*x + 7 = (1/3)*3^2 - 2*3 + 7 this equals 3 - 6 + 7 = 4 --------------------- your vertex should be at the point pair of (x,y) = 3,4) ----------------------- the graph of this equation looks like this: you can see from the graph that the vertex is at the point (x,y) = 3,4)
 Miscellaneous_Word_Problems/388073: 2. LaToya Smith is a buyer for a large manufacturing company . She can purchase 200 pounds of chemicals for \$95. She can also purchase mixing vats for \$400 each. Sales tax on all products is 6% of the total cost of the order. Shipping costs for her order will be a flat \$30. a. At this same rate, how much would 4,000 pounds of chemicals cost? b. This month she doesn’t know yet how many mixing vats she will need to order. Let y = number of mixing vats required. Write an equation using y that she can use to determine the total amount of her order. Assume no chemicals are ordered, only vats. 1 solutions Answer 274433 by Theo(3464)   on 2010-12-20 17:40:35 (Show Source): You can put this solution on YOUR website!200 pounds of chemicals cost \$95. mixing vats = \$400 each. Sales tax on all products is 6% of the total cost of the order. Shipping costs for the order will be a flat \$30. question a: she orders 4000 pounds of chemicals. cost per pound of chemical = \$95.00 / 200 = .475 .475 * 4000 = \$1900.00 for the chemicals. Total cost for the order for 4000 pounds of chemicals only is as follows: Chemicals = \$1900.00 Shipping = \$30.00 Total without Tax = \$1930.00 Tax on \$1930.00 at 6% = \$115.80 Total Cost for the Order = \$1930.00 + \$115.80 = \$2045.80 In equation form, this would be: ((.475 * 4000) + 30) * 1.06 = total cost of the order. Tax is assumed to apply to the total order (material plus shipping). question b: Cost per vat = \$400.00 y = number of vats to order. Total cost of the order for vats only would be: (y * 400 + 30) * 1.06
 logarithm/388101: logx is greater than or equal to log(base 4) x^2 What is x equal to? It tells me to graph it, but i can't figure out how to....1 solutions Answer 274427 by Theo(3464)   on 2010-12-20 16:28:22 (Show Source): You can put this solution on YOUR website!I presume the statement is: log(4,x) >= log(4,x^2) This means that the log of x to the base 4 is greater than or equal to log of x^2 to the base 4. find x. to find where they cross, you need to graph both equations. we'll graph log(4,x) and log(4,x^2) to see if they cross anywhere. from the graph, it looks like this occurs when goes from 0 to 1. when x is negative doesn't apply because the graph of log(4,x) is only valid when x is greater than 0. log(4,0) is not valid. log(4,-x) is not valid. you probably just need to graph them from x = 0 to x = 2. The crossing point is at x = 1. log(4,x) >= log(4,x^2) in the interval 0 < x <= 1 the graph of the log(4,x^2) is still not valid at x = 0. the graph of the log(4,x^2) is valid when x is negative because x^2 is positive.
 Age_Word_Problems/388085: sean is five years older than peter. if the sum of their ages is at most 37 years, what is the olders that peter can be1 solutions Answer 274418 by Theo(3464)   on 2010-12-20 16:05:29 (Show Source): You can put this solution on YOUR website!x = sean's age. x - 5 = peter's age. x + x-5 <= 37 combine like terms to get: 2x - 5 <= 37 add 5 to both sides to get: 2x <= 42 divide both sides by 2 to get: x <= 21 peter's age is x-5 the oldest peter can be is 21-5 = 16 when peter = 16, sean = 16 + 5 = 21 21 + 16 = 37
 Human-and-algebraic-language/388091: how can i make a word problem with 315 divided by 211 solutions Answer 274413 by Theo(3464)   on 2010-12-20 15:59:06 (Show Source): You can put this solution on YOUR website!You have 315 students. There are 21 students per class. How many classes are there? Answer should be 15 classes because 15 * 21 = 315.
 Travel_Word_Problems/388097: Rosario spent \$8.28 for two bags of dog biscuits. If there are 36 biscuits per bag, find the unit cost per biscuits.1 solutions Answer 274412 by Theo(3464)   on 2010-12-20 15:57:22 (Show Source): You can put this solution on YOUR website!He bought 2 bags for \$8.28 There are 36 biscuits in each bag. 2 * 36 = 72 He bought 72 biscuits. Cost per biscuit = \$8.28 / 72 = .115 = 11.5 cents per biscuit. .115 * 72 = \$8.28
 Inequalities/387990: A local gym charges nonmembers \$10 per hour to use the tennis courts. Members pay a yearly fee of \$300 and \$4 per hour for using the tennis courts. Write an equation to find how many hours, h, you must use the tennis courts to justify becoming a member. What is the equation in question #15 for the value of h? Kris, Thank you!1 solutions Answer 274295 by Theo(3464)   on 2010-12-20 04:38:22 (Show Source): You can put this solution on YOUR website!The formula for total cost is: T = x + y*h x = the membership cost. y = the cost per hour. For the non-member, the total cost is equal to: T = 0 + 10*h which becomes: T = 10*h For the member, the total cost is equal to: T = 300 + 4*h You want to know at what point the total cost for the member becomes less than the total cost for the non-member. That point is given by the equation: 300 + 4*h < 10*h Subtract 4*h from both sides of this equation to get: 300 < 10*h - 4*h Simplify to get: 300 < 6*h Divide both sides of this equation by 6 to get: 300/6 < h Simplify to get: 50 < h 50 < h also means h > 50. The total membership cost will becomes less than the total non-member cost when the number of hours is greater than 50. If h = 50, then the equation of: 300 < 10*h - 4*h becomes: 300 < 10*50 - 4*50 Simplify this to get: 300 < 500 - 200 Simplify further to get: 300 < 300 The statement is not true because h is not greater than 50. Take any h greater than 50 and the equation should be true. Take h = 51. You get: 300 < 10*51 - 4*51 simplify this to get: 300 < 510 - 204 Simplify further to get: 300 < 306 The statement is now true.
 Percentage-and-ratio-word-problems/387998: 1/4 of this seniors have averages above 90. 1/2 of the remainders have averages between 80 and 90. what part of the senior class have averages below 80?1 solutions Answer 274294 by Theo(3464)   on 2010-12-20 04:26:26 (Show Source): You can put this solution on YOUR website!1/4 of the seniors have averages above 90. The remainder is 3/4 of the seniors. 1/2 of the remainders have averages between 80 and 90. That means that 1/2 * 3/4 of the seniors have averages between 80 and 90. That means that 3/8 of the seniors have averages between 80 and 90. The total number of seniors that have averages above 80 is equal to 1/4 + 3/8. Since 1/4 = 2/8, then the total number of seniors that have averages above 80 is equal to 5/8. That leaves 3/8 of the seniors that have averages below 80 because 5/8 + 3/8 = 8/8 = 1.
 Rate-of-work-word-problems/388002: 1. My husband is a terrific cleaner. If he cleans the living room, it takes him 2 hours and 5 minutes. If I help, we can do the job in 1 hour and 20 minutes. How long does it take for me to do it by myself? 1 solutions Answer 274293 by Theo(3464)   on 2010-12-20 04:20:43 (Show Source): You can put this solution on YOUR website!Rate * Time = Units Produced R*T=U The number of units produced is 1 clean house. U = 1 Working together, the rate of each individual is additive. R1 = his rate R2 = your rate (R1 + R2) * T = 1 He takes 2 hours and 5 minutes to clean the house by himself. 5 minutes = 5/60 hours = 1/12 hours = .0833333333 hours. He take 2.083333333 hours to clean the house by himself. Working by himself, the formula becomes R1*T = U which becomes: R1*2.08333333 = 1 Divide both sides of this equation by 2.0833333, and you get: R1 = 1.2.0833333 = .48 his rate is equal to .48 of the house in 1 hour. The formula for both of you cleaning the house together becomes: (.48 + R2) * T = 1 The time it takes both of you to clean the house is 1 hours and 20 minutes. 20 minutes = 20/60 hours = .3333333 hours. The time it takes both of you to clean the house is 1.3333333 hours. The formula becomes: (.48 + R2) * 1.3333333 = 1 Remove parentheses to get: .48 * 1.33333 + R2 * 1.3333333 = 1 Simplify to get: .64 + R2 * 1.3333333 = 1 Subtract .64 from both sides of the equation and then divide both sides of the equation by 1.3333333 to get: R2 = (1 - .64) / 1.3333333 Simplify to get: R2 = .36 / 1.3333333 which becomes: R2 = .27 Your rate of cleaning the house is equal to .27 of the house in 1 hour. The formula for both of you completing the house at the same time becomes: (.48 + .27) * 1.33333333 = 1 Simplify this to get: .75 * 1.3333333 = 1 Simplify further to get: 1 = 1 The formula is good. You can clean .27 of the house in 1 hour. The formula for you working alone becomes: R2 * T = 1 Since R2 = .27, this becomes: .27 * T = 1 Divide both sides of this equation by .27 to get: T = 3.703703704 house. You would take 3.703703704 hours to clean the house by yourself. .703703704 hours * 60 = 42.222222 minutes. If would take approximately 3 hours and 42 minutes for you to clean the house by yourself at that rate.
 Miscellaneous_Word_Problems/388003: Use inductive reasoning to predict the next two terms in the pattern. Justify your prediction 1, 8, 27, 64, 125...1 solutions Answer 274291 by Theo(3464)   on 2010-12-20 03:52:45 (Show Source): You can put this solution on YOUR website!1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125 6^3 = 216 The next term is 216. Every term is equal to the next number in the sequence cubed.
 Graphs/387550: Find the slope and the y-intercept of each line whose equation is given. y=-3/8x+4 PLEASE HELP ME..!!! 1 solutions Answer 273986 by Theo(3464)   on 2010-12-18 09:56:16 (Show Source): You can put this solution on YOUR website!slope intercept form of the equation of a straight line is y = mx + b m is the slope b is the y intercept your equation is y = (-3/8)*x + 4 this equation is already in slope intercept form. the slope is (-3/8) the y intercept is 4
 Expressions-with-variables/387557: I really need HELP! Week 4 Application Assignment Answer the following questions. In 2004 Major League Baseball, Barry Bonds, playing for the San Francisco Giants of the National League, collected 44 hits in 117 at-bats in his first 49 games. The ratio of number of hits to number of at-bats, rounded to the nearest thousandth, is a player’s batting average. What was Bond’s batting average in his first 49 games? Set the ratio hits to at-bats and convert to a decimal. Based on the ratio of number of hits to number of games, how many hits would he get in a 162-game season? Set a proportion and solve for the unknown. Based on the ratio of number of hits to number of at-bats and assuming he bats 400 times in 2004, how many hits would he get? 1 solutions Answer 273985 by Theo(3464)   on 2010-12-18 09:51:48 (Show Source): You can put this solution on YOUR website!he got 44 hits in 117 at bats in 49 games. ratio of hits to at bats is equal to 44/117 44/117 = .376068376. rounded to the nearest thousandth that equals .376 ratio of hits to games is equal to 44/49. 44/49 = .897959184 based on the ratio of hits to games, he will get .897959184 * 162 = 145.4693878 hits in a 162 game season. based on the ratio of hits to at bats, he will get .376068376 * 400 = 150.4273504 hits in 400 at bats. these round to 145 hits in a 162 game season and 150 hits in 400 at bats.
 Graphs/387552: Find the slope and the y-intercept of each line whose equation is given. m = −3; (−2,1) SOMEONE PLEASE HELP !!!!!!!!!!!1 solutions Answer 273983 by Theo(3464)   on 2010-12-18 09:29:05 (Show Source): You can put this solution on YOUR website!m is the slope and the line is passing through the point (x,y) = (-2,1) This means that x = -2 and y = 1 at that point on the graph. the slope intercept form of the equation of a straight line is y = m*x + b m is the slope b is the y-intercept replace m with -3 to get: y = -3x + b replace y with 1 and x with -2 and then solve for b. you get: 1 = -3*(-2) + b this becomes: 1 = 6 + b this results in b = -5 your equation becomes: y = -3x - 5 graph of your equation looks like this: you can see that the graph passes through the point (-2,1) when x = -2, y = -3*(-2) - 5 = 6 - 5 = 1 since the slope is -3, then for every + 1 unit change in the value of x, the value of y should change by - 3 units. when x = 0, y = -5. when x = 1, y = -8. when x = 1, y = -11. etc.
 Polynomials-and-rational-expressions/387337: I am lost on this problem: (35b^(3)+25b^(2)+30b+47)÷(5b+5) I started with the first part (35b^(3)+25b^(2)+30b+47)and I tried to divide each part by the second part (5b) (5b) (5b) (5) but none of my answers were the same as the possible answers. 1 solutions Answer 273975 by Theo(3464)   on 2010-12-18 07:53:14 (Show Source): You can put this solution on YOUR website!I get an answer of 7b^2 - 2b + 8 first you divide 35b^3 by 5b to get 7b^2 then you multiply (5b+5) by 7b^2 to get (35b^3 + 35b^2) then you subtract (35b^3 + 35b^2) from (35b^3 + 25b^2) to get -10b^2 then you bring down the 30b to get (-10b^2 + 30b) then you divide -10b^2 by 5b to get -2b then you multiply (5b+5) by -2b to get (-10b^2 - 10b) then you subtract (-10b^2 - 10b) from (-10b^2 + 30b) to get 40b then you bring down the 47 to get (40b + 47) then you divide 40b by 5b to get 8 then you multiply (5b+5) by 8 to get (40b + 40) then you subtract (40b + 40) from (40b + 47) to get 7 that's your remainder. the answer is 7b^2 - 2b + 8 with a remainder of 7 on paper, it looks like what is shown below.
 Polynomials-and-rational-expressions/387519: I need help dividing (x^2+6x-50)/(x+8)1 solutions Answer 273972 by Theo(3464)   on 2010-12-18 07:23:50 (Show Source): You can put this solution on YOUR website!Answer should be: (x-2) with a remainder of -34 remainder of -34 is equivalent to -34/(x+8) answer is therefore x-2 - 34/(x+8) the process works like this. divide x^2 by x to get x because x*x = x^2 multiply (x+8) by x to get x^2 + 8x subtract (x^2 + 8x) from (x^2 + 6x) to get -2x bring down the -50 to get (-2x - 50) divide -2x by x to get -2 multiply (x+8) by -2 to get (-2x - 16) subtract (-2x -16) from (-2x - 50) to get -34 that's your remainder. your answer is (x-2) with a remainder of -34 when you multiply (x-2) by (x+8) you will then need to add the remainder to get back to the original equation. (x-2) * (x+8) = x^2 + 6x - 16 add -34 to that and you get x^2 + 6x - 50 which was your original equation. on paper, the division will be as shown below:
 Inequalities/384715: Find the greatest possible pair of integers such that one integer is twice the other and their sum is less than or equal to 12.1 solutions Answer 272344 by Theo(3464)   on 2010-12-12 06:33:13 (Show Source): You can put this solution on YOUR website!x = one of the integers y = the other x = 2*y x + y <= 12 substitute for x in the second equation to get: 2*y + y <= 12 combine like terms to get: 3*y <= 12 divide both sides by 3 to get: y <= 4 the greatest that y can be is 4. if y is equal to 4, then x = 2*y means that x = 8 the greatest that x can be is 8. when x = 8 and y = 4, then: 8 = 2*4 = ok 4 + 8 = 12 = ok
 Equations/384661: ***I NEED THE STEPS TO SOLVE THIS PROBLEM SO THAT I CAN BETTER UNDERSTAND IT...PLEASE HELP*** ANSWER: (a)2sqroot2 ft (b) 2ft PROBLEM: DIMENSIONS OF AN AQUARIUM An open-topped aquarium is to be constructed with 6ft-long sides and square ends. (a) Find the height of the aquarium if the volume is to be 48ft^3. (b) Find the height if 44ft^2 of glass is to be used. THANK YOU!!!!!!!!1 solutions Answer 272343 by Theo(3464)   on 2010-12-12 06:22:05 (Show Source): You can put this solution on YOUR website!6 foot long sides. square ends. find height if volume is 48 cubic feet Volume of rectangular prism is h*l*w (height * length * width) length is 6 feet. height and width (form the ends) are the same measure, call it s. volume is therefore h*s^2 (height * area of the base) formula for volume is v = h*s^2 height is 6 feet, so formula becomes v = 6*s^2 volume is 48 cubic feet, so formula becomes 48 = 6*s^2 divide both sides of equation by 6 to get 8 = s^2 take square root of both sides of equation to get s = sqrt(8). length is 6 width is sqrt(8) height is sqrt(8) volume is l*w*h = 6*sqrt(8)*sqrt(8) = 6*8 = 48 cubic feet. sqrt(8) = sqrt(2*2*2) = 2*sqrt(2) -------------------------------------- (b) Find the height if 44 square feet of glass is to be used. aquarium is open topped. length is still 6 feet. ends are still square. aquarium is open on top. it therefore has 2 sides (one on each end) and 3 sides along the length. since the ends are still square, then the width equal the height and we can still call them s rather than w and h. the area of each end is still s^2. 2 times the area of each end is therefore equal to 2*s^2. the area of each side along the length will be s*l. we have 3 of them. the total area of the sides along the length is therefore 3*s*l the side along the length on the top of the aquarium is open and so its area is not counted for total area of glass. total area of glass is the area of the 2 ends plus the area along the length for 3 of the 4 sides along the length.. that equals 2*s^2 + 3*s*l which equals 44 square feet. formula becomes 44 = 2*s^2 + 3*s*l we know that l = 6, so we substitute 6 for l to get: 44 = 2*s^2 + 3*s*6 which becomes 44 = 2*s^2 + 18*s subtract 44 from both sides of this equation to get the standard form of a quadratic equation of: 2*s^2 + 18*s -44 = 0 divide both sides of this equation by 2 to get: s^2 + 9*s - 22 = 0 factor this equation to get: (s+11) * (s-2) = 0 solve for s to get s = -11 or s = 2 s can't be negative, so s has to be equal to 2. that's your answer. total surface area = 2*4 + 3*2*6 = 8 + 36 = 44 square inches. a picture of your aquarium that might help is shown here:
 Polynomials-and-rational-expressions/384398: How can you have one solution in something that is quadratic?1 solutions Answer 272130 by Theo(3464)   on 2010-12-11 08:15:21 (Show Source): You can put this solution on YOUR website!When b^2 - 4ac is equal to 0, you will have 1 solution. The standard form of the quadratic equation is: ax^2 + bx + c = 0 a is the coefficient of the x^2 term. b is the coefficient of the x term. c is the constant term. To find the roots of this equation, you can use the quadratic formula. the quadratic formula is: ``` x = -b +/- sqrt(b^2-4ac) -------------------- 2a ``` If b^2 - 4ac is equal to 0, then you have +/- 0 in the numerator and there is only one solution to the problem. We can construct an example to show you what I mean. To construct this example, let's pick a b term at random. Let's say b is equal to 6. That makes b^2 = 36 Divide 36 by 4 to get 9. We need to get a * c to be equal to 9. a * c will be equal to 9 when a = 1 and c = 9 We have: a = 1 b = 6 c = 9 b^2 - 4ac becomes 36 - (4*1*9) which becomes 36 - 36 which becomes 0. Our quadratic equation should have one solution only. The standard form of the quadratic equation is ax^2 + bx + c = 0 Since a = 1, b = 6 and c = 9, this equation becomes: x^2 + 6x + 9 = 0 We will now graph this equation as shown below: You can see from the graph that the graph of the quadratic equation touches the x-axis at the point x = -3 and only at that point.