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# Recent problems solved by 'AnlytcPhil'

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 Rational-functions/43914: This question is from textbook Algebra and Trigonometry with Analytic Geometey The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3+i among its roots. Express f(x) as a product of linear and quadratic polynomials polynomials with real coefficients. This question I'm having trouble with but I came up witha an answer of; f(x)=(x-4)(x^2-6x-9) is this right. thanks for looking at this question.1 solutions Answer 28812 by AnlytcPhil(1277)   on 2006-06-27 05:49:44 (Show Source): You can put this solution on YOUR website!```The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3+i among its roots. Express f(x) as a product of linear and quadratic polynomials polynomials with real coefficients. This question I'm having trouble with but I came up witha an answer of: f(x)=(x-4)(x^2-6x-9) is this right. thanks for looking at this question. No that's not correct. You must know the following facts about polynomials: 1. A degree n polynomial has n roots, counting multiplicities. 2. If r is a root of a polynomial then (x-r) is a factor of the polynomial. 3. If a polynomial with real coefficients has p+qi as one root, it also has its conjugate p-qi as another root. A, By 2 above, since 4 is a root then (x-4) is a factor. (You did this) B. By 2 above, since 3+i is a root then [x-(3+i)] is a factor of the polynomial. C. By 3 above, since 3+i is a root then 3-1 is a factor of the polynomial. D. By C and 2, since 3-i is a root then [x-(3-i)] is a factor of the polynomial. E. By 1, there are no more roots besides 4, 3+i, and 3-1 Therefore f(x) is the product of the three factors. It will have leading coefficient 1 because the c oefficients of x in all three factors is 1. [Note: If it had had a leading coefficient other than 1, we would have to multiply the polynomial by it, too, but thet is unnecessary here.] So we have: f(x) = (x - 4)[x - (3+i)][x - (3-i)] But the two factors containing imaginary numbers have to be multiplied together, since all coefficients must be real. We can use FOIL to multiply the last two factors together: f(x) = (x - 4)[x - (3+i)][x - (3-i)] f(x) = (x - 4)[x² - (3-i)x - (3+i)x + (3+i)(3-i)] f(x) = (x - 4)[x² - (3x-ix) - (3x+ix) + (9-3i+3i-i²)] f(x) = (x - 4)[x² - 3x + ix - 3x - ix + (9-i²)] f(x) = (x - 4)[x² - 6x + 9 - i²] Now since i² = -1 we substitute (-1) for i². f(x) = (x - 4)[x² - 6x + 9 - (-1)] f(x) = (x - 4)[x² - 6x + 9 + 1] f(x) = (x - 4)[x² - 6x + 10] f(x) = (x - 4)(x² - 6x + 10) Edwin AnlytcPhil@aol.com```
 Rational-functions/43691: This question is from textbook INTERMEDIATE ALGEBRA ```rationalize the denominator: 7 --------- √3 – √2```1 solutions Answer 28685 by AnlytcPhil(1277)   on 2006-06-25 20:25:04 (Show Source): You can put this solution on YOUR website!``` 7 --------- √3 – √2 To rationalize a denominator with has two terms, you must 1. Form the conjugate surd A. Write the denominator √3 - √2 B. Change the sign of the second term only √3 + √2 2. Form the fraction by putting the conjugate surd over itself. √3 + √2 --------- (Notice that this equals the number 1] √3 + √2 3. Multiply the original fraction by this fraction. The value won't change because this amounts to multiplying by 1. 7 √3 + √2 --------- × --------- √3 – √2 √3 + √2 4. Indicate the multiplication of numerators ans denominators 7(√3 + √2) ------------------- (√3 – √2)(√3 + √2) 5. Multiply the top and the bottom 7√3 + 7√2 ------------------------ 3 + √3√2 - √2√3 - 2 6. The middle two terms in the bottom cancel out 7√3 + 7√2 ----------- 3 - 2 7√3 + 7√2 ----------- 1 7√3 + 7√2 Edwin AnlytcPhil@aol.com```
 Graphs/43667: Give the equation for the circle with center C(3,-2) and radius 4. Thanks!1 solutions Answer 28660 by AnlytcPhil(1277)   on 2006-06-25 16:19:15 (Show Source): You can put this solution on YOUR website!```Give the equation for the circle with center C(3,-2) and radius 4. Thanks! You just have to learn the equation (x-h)² + (y-k)² = r² Where the center = (h,k) = (3,-2) Substitute 3 for h and -2 for k and 4 for the radius r. Then you have (x-3)² + (y-(-2))² = 4² You can simplify (x-3)² + (y+2)² = 16 And you can multiply it out if you like (x-3)(x-3) + (y+2)(y+2) = 16 x² - 3x + 9 + y² + 4y + 4 = 16 x² + y² - 3x + 4y + 13 = 16 x² + y² - 3x + 4y - 3 = 0 Edwin```
 Points-lines-and-rays/43501: This question is from textbook Algebra and trigonometry with analytic geometry If P(4,-5) is a point on the graph of the function y=f(x), find the corresponding point on the graph of y=2f9x-6). Not sure but I think it is (6,8). Thanks for looking it over. 1 solutions Answer 28504 by AnlytcPhil(1277)   on 2006-06-24 07:18:46 (Show Source): You can put this solution on YOUR website!``` If P(4,-5) is a point on the graph of the function y=f(x), find the corresponding point on the graph of y = 2f(x-6). Not sure but I think it is (6,8). Thanks for looking it over. I'm afraid that's wrong. y = 2f(x-6) This requires two operations of y = f(x) 1. First we go from y = f(x) to y = f(x-6) That shifts the graph 6 units right. So the point (4,-5) moves right to the point (10, -5) on the graph of y = f(x-6) 2. Then we go from y = f(x-6) to y = 2f(x-6) That stretches the graph double vertically. Imagine the graph drawn on a rubber sheet and stretched double vertically. The points above the x-axis will stretch twice as high and the points below the x-axis will stretch twice as low. Therefore the point (10. -5), being below the x-axis, will stretch twice as low or down to (10, -10) Edwin AnlytcPhil@aol.com```
 Miscellaneous_Word_Problems/43505: This question is from textbook Algebra and trigonometry with analytic geometry An object is projected upward from the top of a water tower. Its distance in feet above the ground after t seconds is given by s(t)=-16t^2+64t+80. How many seconds will it take to reach ground level? I came out with 4 seconds after working it out. thanks for checking.1 solutions Answer 28502 by AnlytcPhil(1277)   on 2006-06-24 06:13:31 (Show Source): You can put this solution on YOUR website!``` An object is projected upward from the top of a water tower. Its distance in feet above the ground after t seconds is given by s(t) = -16t^2 + 64t + 80. How many seconds will it take to reach ground level? I came out with 4 seconds after working it out. thanks for checking. Let's check your answer and see if it is correct: We substitute 4 seconds into the equation: s(t) = -16t² + 64t + 80 s(4) = -16(4)² + 64(4) + 80 s(4) = -16(16) + 256 + 80 s(4) = -256 + 256 + 80 s(4) = 80 feet above ground After 4 seconds the object is still 80 feet above the ground. So 4 seconds can't be right. Let's do it right: Ground level is ground zero, that is, when the height above the ground s(t) is zero, So replace s(t) by 0 s(t) = -16t² + 64t + 80 0 = -16t² + 64t + 80 16t² - 64t - 80 = 0 Divide every term through by 16 t² - 4t - 5 = 0 (t - 5)(t + 1) = 0 t - 5 = 0; t + 1 = 0 t = 5 t = -1 Answer: 5 seconds. Discard the negative answer. It takes 5 seconds to reach the ground. The water tank is 80 feet high. At the instant when 2 seconds have passed it reaches its maximum height of 144 feet (64 feet above the water tower). Then it starts falling back down. At the instant when 4 seconds have passed, it is back even with the tower. Then after 5 seconds have passed the object reaches the ground. Edwin AnlytcPhil@aol.com```
 Linear_Algebra/43437: For the set of data below, find the line of best fit. x -2 -1 0 1 2 y 7 6 3 2 0 a. y = -1.8x + 3.6 b. y = -0.54x + 1.95 c. y = 3.6x - 1.8 d. y = 1.95x - 0.54 e. None of the others 1 solutions Answer 28450 by AnlytcPhil(1277)   on 2006-06-23 16:58:45 (Show Source): You can put this solution on YOUR website!``` For the set of data below, find the line of best fit. x -2 -1 0 1 2 y 7 6 3 2 0 a. y = -1.8x + 3.6 b. y = -0.54x + 1.95 c. y = 3.6x - 1.8 d. y = 1.95x - 0.54 e. None of the others Do you have a TI-82 or higher? Press STAT ENTER Put the 5 x-values under L1 Put the 5 y-values under L2 Press STAT then RIGHT ARROW to highlight CALC Scroll down to LinReg(ax+b) Press ENTER Read y=ax+b a=-1.8 b=3.6 ··· So substitute -1.8 for a and 3.6 for b in y = ax+b y = -1.8x + 3.6 So it's choice 1. Were you supposed to use a formula? Edwin```
 Equations/43386: If f(x)=5x-1, find the following f(a-2)1 solutions Answer 28397 by AnlytcPhil(1277)   on 2006-06-23 09:06:20 (Show Source): You can put this solution on YOUR website!```If f(x) = 5x - 1, find the following f(a-2) f(a number or expression) = right hand side just means to substitute that number or expression for x in the right hand side and then simplify it: f(x) = 5x - 1 Erase all the x's and replace them with blanks f___ = 5_ - 1 Make the blanks long enough to hold (a-2) f_____ = 5_____ - 1 Put (a-2)'s in the blanks. f(a-2) = 5(a-2) - 1 Erase the blanks: f(a-2) = 5(a-2) - 1 Simplify the right side: f(a-2) = 5a - 10 - 1 f(a-2) = 5a - 11 Edwin```
 logarithm/43262: solve x logx 25 = -2 ***the x is subscript of log1 solutions Answer 28295 by AnlytcPhil(1277)   on 2006-06-22 12:23:59 (Show Source): You can put this solution on YOUR website!```solve for x logx 25 = -2 Use the rule that says: A logarithmic equation of the form logBA = C is equivalent to and can be rewritten as the exponential equation BC = A So we rewrite logx25 = -2 as x-2 = 25 Then we write the x-2 as 1/x2 1/x2 = 25 Multiply both sides by x2: 1 = 25x2 Divide both sides by 25 1/25 = x2 Take positive square roots of both sides (logarithm bases must be positive and ¹ 1, by definition) 1/5 = x Edwin```
 Quadratic-relations-and-conic-sections/43230: This question is from textbook algebra and trigonometry with analytic geometry Give the equation for the right half of the circle. C(2,3). I plotted it out to x=2+squrt9-(y-3)^2. Not sure about the signs if I have them right or not. thanks for checking. 1 solutions Answer 28251 by AnlytcPhil(1277)   on 2006-06-22 09:06:53 (Show Source): You can put this solution on YOUR website!```Give the equation for the right half of the circle. C(2,3). I plotted it out to x=2+squrt9-(y-3)^2. Not sure about the signs if I have them right or not. thanks for checking. I will assume the radius is 3. You forgot to give that. The equation of the circle is (x - 2)² + (y - 3)² = 3² (x - 2)² + (y - 3)² = 9 Since you want the right half, solve for x. (x - 2)² = 9 - (y - 3)² Since you want the right half, take positive square roots of both sides: ____________ x - 2 = Ö9 - (y - 3)² ____________ x = 2 + Ö9 - (y - 3)² Yes it looks like you're right. Edwin```
 Linear_Equations_And_Systems_Word_Problems/43220: A bus leaves a station at 1 p.m., traveling west at an average rate of 44 mi/h. One hour later a second bus leaves along the same route, traveling at 54 mi/h. At what time will the two buses be 274 mi apart?1 solutions Answer 28248 by AnlytcPhil(1277)   on 2006-06-22 08:39:10 (Show Source): You can put this solution on YOUR website!```A bus leaves a station at 1 p.m., traveling west at an average rate of 44 mi/h. One hour later a second bus leaves along the same route, traveling at 54 mi/h. At what time will the two buses be 274 mi apart? Wow!!!!!! That'll sure take a long time with the second bus traveling only 10 mi/hr faster than the first bus!!!!!!! At only 10 mi/hr faster, the second bus won't even catch up to the first to pass it until 4.4 hours later at 6:24 PM. Then to get 274 miles ahead will take more than a whole day. Are you sure you have this one copied right? It's a ridiculous problem, but it can be imagined. So I'll do it anyway. Let's assume they started on Monday. It will be at least Tuesday evening before the 2nd bus is 274 miles ahead of the first bus, going only 10 mph faster. Make this chart D R T 1st bus| 2nd bus| Let t be the time the first bus traveled. Then since the 2nd bus started an hour later its traveling time is t-1. So fill in the times D R T 1st bus| t 2nd bus| t-1 Fill in the given rates (speeds): D R T 1st bus| 44 t 2nd bus| 54 t-1 Use D = RT to fill in the distances: D R T 1st bus| 44t 44 t 2nd bus| 54(t-1) 54 t-1 Now the distance the 2nd bus traveled is 274 miles more than the 1st bus So we form the equation by 2nd bus's distance = 1st bus's distance + 274 54(t-1) = 44t + 274 Answer = 32.8 hours So if the first bus started at 1 p.m on Monday the second bus will be 274 miles ahead of the 1st bus at 9:48 p.m Tuesday night. I think you copied the problem wrong. Edwin```
 Quadratic-relations-and-conic-sections/43224: This question is from textbook algebra and trigonometry with analytic geometry Give the equation for the circle with center C(3,-2) and radius 4. I came out with (x-3)^2+(y-2)^2=16. But I'm not sure if I worked it our right. thanks for looking it over.1 solutions Answer 28244 by AnlytcPhil(1277)   on 2006-06-22 07:19:55 (Show Source): You can put this solution on YOUR website!```Give the equation for the circle with center C(3,-2) and radius 4. I came out with (x-3)^2+(y-2)^2=16. But I'm not sure if I worked it our right. thanks for looking it over. You missed one sign (x - h)² + (y - k)² = r² h = 3, k = -2 [x - (3)]² + [y - (-2)]² = (4)² [x - 3]² + [y + 2]² = 16 (x - 3)² + (y + 2)² = 16 To do it quickly just remember to change both signs of the coordinates of the center. Edwin```
 Unit_Conversion_Word_Problems/43228: A recipe calls for 1 1/3 tablespoons of butter and serves 6 people.If you wish to cook this recip for yourself,how much butter should you use?1 solutions Answer 28243 by AnlytcPhil(1277)   on 2006-06-22 07:07:43 (Show Source): You can put this solution on YOUR website!```A recipe calls for 1 1/3 tablespoons of butter and serves 6 people.If you wish to cook this recipe for yourself, how much butter should you use? You would divide 1 1/3 by 6. (1 1/3) ÷ 6 = (4/3) ÷ (6/1) (4/3) × (1/6) 4/18 2/9 tablespoons Since there are no measuring spoons for ninths, we use the fact that 1 tablespoon = 3 teaspoons 2/9 tablespoon = (2/9) × (3 teaspoons) = (2/9)×(3/1) teaspoon = 6/9 teaspoon = 2/3 teaspoon There are no third teaspoon measuring spoons either so you have to measure a teaspoon and take a litle bit out. Edwin```
 Rational-functions/43227: This question is from textbook INTERMEDIATE ALGEBRA ```Divide and Simplify: 5x - 7 ----------- 25x² - 49 ---------------- 5x² + 27x + 28```1 solutions Answer 28242 by AnlytcPhil(1277)   on 2006-06-22 06:53:20 (Show Source): You can put this solution on YOUR website!```Divide and Simplify: 5x - 7 ----------- 25x² - 49 ---------------- 5x² + 27x + 28 Write it as the upper fraction divided by the the trinomial at the bottom: 5x - 7 ----------- ÷ (5x² + 27x + 28) 25x² - 49 Now put that trinomial over 1: 5x - 7 5x² + 27x + 28 ----------- ÷ ---------------- 25x² - 49 1 Now invert the second fraction and change the ÷ to × 5x - 7 1 ----------- × ---------------- 25x² - 49 5x² + 27x + 28 Factor the two denominators 5x - 7 1 ------------------ × ----------------- (5x - 7)(5x + 7) (x + 4)(5x + 7) Write as one fraction by indicating the multiplicatiom of numerators and denominators: 5x - 7 --------------------------------- (5x - 7)(5x + 7)(x + 4)(5x + 7) Cancel the (5x - 7)'s 1 5x - 7 --------------------------------- (5x - 7)(5x + 7)(x + 4)(5x + 7) 1 1 ------------------------- (5x + 7)(x + 4)(5x + 7) Since there are two factors (5x + 7) you can write their product as (5x + 7)² 1 ------------------ (x + 4)(5x + 7)² Edwin```
 Graphs/43164: Evaluate h(-2/3) if h(x) =[[x-2]] I get [[-2/3-2]]. Is the next step[[ -22/3]] or [[-1-2]]? Please help Thanks1 solutions Answer 28181 by AnlytcPhil(1277)   on 2006-06-21 20:04:25 (Show Source): You can put this solution on YOUR website!```Evaluate h(-2/3) if h(x) =[[x-2]] Double brackets indicates the "floor function". It means the largest integer that does not exceed the number between the double brackets: h(x) =[[x-2]] h(-2/3) = [[-2/3-2]] h(-2/3) = [[-2/3 - 6/3]] h(-2/3) = [[-8/3]] h(-2/3) = [[-2.888888888888···]] The largest integer that does not exceed -2.888888888888··· is -3. So h(-2/3) = -3 Edwin AnlytcPhil@aol.com```
 Polynomials-and-rational-expressions/43121: This question is from textbook Beginning Algebra One number is 5 times another...the sum of their reciprocals is 6/35...what are the 2 numbers? Um...what? Please help...1 solutions Answer 28154 by AnlytcPhil(1277)   on 2006-06-21 16:11:10 (Show Source): You can put this solution on YOUR website!```One number is 5 times another...the sum of their reciprocals is 6/35...what are the 2 numbers? Smaller number = x >>...one number is 5 times another...<< Larger number = 5x Reciprocal of smaller number = 1/x Reciprocal of larger number = 1/(5x) >>...the sum of their reciprocals is 6/35...<< 1/x + 1/(5x) = 6/35 Can you solve that? If not post again. Hint: multiply thru by LCD = 35x Answer: x = 7, So smaller number = x = 7 Larger number = 5x = 5(7) = 35 Edwin```
 Trigonometry-basics/43113: The approach pattern to an airport requires pilots to set an 11o angle of descent toward a runway. If a plane is flying at an altitude of 9,500 m, at what distance (measured along the ground) from the airport must the pilot decend 1 solutions Answer 28148 by AnlytcPhil(1277)   on 2006-06-21 16:02:45 (Show Source): You can put this solution on YOUR website!``` The approach pattern to an airport requires pilots to set an 11o angle of descent toward a runway. If a plane is flying at an altitude of 9,500 m, at what distance (measured along the ground) from the airport must the pilot decend P · · | 9500 m A · 11° | ·--------------------------------- x B The plane is at P, the Airport is at A, B is a point on the ground directly under the plane. We want to find AB, the length of which we call x 9500 is the opposite side to 11° x is the adjacent side to 11° tan(11°) = opposite/adjacent tan(11°) = 9500/x x·tan(11°) = 9500 x = 9500/tan(11°) x = 48873.26315 m or about 48900 meters. Edwin AnlytcPhil@aol.com```
 absolute-value/43104: Can anyone solve the following? 27< 4|y + 9|-9 Thanks so much1 solutions Answer 28131 by AnlytcPhil(1277)   on 2006-06-21 15:35:50 (Show Source): You can put this solution on YOUR website!```Can anyone solve the following? 27 < 4|y + 9| - 9 You must learn two principles for inequalities to remove the absolute value bars: ----------------------------------- 1. If C > 0 |Ax + B| < C can be rewritten as " -C < Ax + B < C " ----------------------------------- 2. If C > 0 |Ax + B| > C can be rewritten as " Ax + B < -C OR Ax + B > C " ------------------------------------ Note the above two principles also hold for < and > -4|y + 9| + 27 < -9 -4|y + 9| < -9 - 27 -4|y + 9| < -36 Divide both sides by -4, which reverses the inequality: |y + 9| > 9 This is principle 2 above y + 9 < -9 OR y + 9 > 9 y < -18 OR y > 0 Plot on a numberline: <======o-----------o======> -18 0 Interval notation: (-¥, -18) È (0, ¥) Edwin AnlytcPhil@aol.com```
 Trigonometry-basics/43108: Choose the correct function value for: csc 23°20'1 solutions Answer 28123 by AnlytcPhil(1277)   on 2006-06-21 15:11:17 (Show Source): You can put this solution on YOUR website!```Choose the correct function value for: csc 23°20' csc(23°20') = 1/sin(23°20') If you have a TI-82 or better calculator Press MODE to make sure "Degree" mode is higlighted, not "Radian" To get ° press 2nd ANGLE 1 To get ' press 2nd ANGLE 2 Answer 2.524743968 Edwin AnlytcPhil@aol.com```
 Quadratic_Equations/43105: ```Solve the following systems: 2x - y = 1 6y - 2x = 9```1 solutions Answer 28118 by AnlytcPhil(1277)   on 2006-06-21 15:01:08 (Show Source): You can put this solution on YOUR website!```Solve the following systems: 2x - y = 1 6y - 2x = 9 By substitution: Solve the first for y: 2x - y = 1 2x - 1 = y y = 2x - 1 Substitute (2x - 1) for y in the second: 6y - 2x = 9 6(2x - 1) - 2x = 9 12x - 6 - 2x = 9 10x - 6 = 9 10x = 15 x = 15/10 x = 3/2 Substitute 3/2 for x in y = 2x - 1 y = 2x - 1 y = 2(3/2) - 1 y = 3 - 1 y = 2 The solution is (x,y) = (3/2, 2) Edwin AnlytcPhil@aol.com ```
 logarithm/43039: ```Given that 0.475 < log base 10 of 3 < 0.478, how many digits are in the number 350? 1 solutions Answer 28065 by AnlytcPhil(1277)   on 2006-06-21 06:20:08 (Show Source): You can put this solution on YOUR website!Given that 0.475 < log base 10 of 3 < 0.478, how many digits are in the number 350? We have to get 350 into the form 10x We need to find x such that 350 = 10x Take logs (base 10 understood) of both sides log(350) = log(10x) Using rules of logarithms: 50·log(3) = x Solve for log(3) log(3) = x/50 We are given that 0.475 < log(3) < 0.478 So substitute x/50 for log(3) 0.475 < x/50 < 0.478 Multiply thru by 50 23.75 < x < 23.9 Raise 10 to all three powers: 1023.75 < 10x < 1023.9 1023.75 < 350 < 1023.9 Integers equal to or greater than 100 but less than 101 have 1 digit. Integers equal to or greater than 101 but less than 102 have 2 digits. Integers equal to or greater than 102 but less than 103 have 3 digits. Integers equal to or greater than 103 but less than 104 have 4 digits. ... ... ... ... ... ... ... ... Integers equal to or greater than 1023 but less than 1024 have 24 digits. 1023 < 1023.75 < 350 < 1023.9 < 1024 Thus 350 has 24 digits. Checking with the calculator: 350 = 7.176879877×1023 Edwin AnlytcPhil@aol.com ```
 Equations/43042: I know I make common mistakes when solving equations can you help with the following: The equation: 2(x -2) =x+3 First step in solving: 2x-2 =x+3 Can you tell me a clear explanation of what error has been made. What could be done to avoid this error? 1 solutions Answer 28064 by AnlytcPhil(1277)   on 2006-06-21 05:15:29 (Show Source): You can put this solution on YOUR website!```I know I make common mistakes when solving equations can you help with the following: The equation: 2(x -2) =x+3 First step in solving: 2x-2 =x+3 Can you tell me a clear explanation of what error has been made. What could be done to avoid this error? Your error is in the distributive law or principle. You have 2(x - 2) = x + 3 Then you have 2x - 2 = x + 3 That is wrong. It should be 2x - 4 = x + 3 That's because when you have 2(x - 2) you multiply the red 2 first times the x, which you did, but ALSO you must multiply the red 2 times the blue - 2 as well. What you did was to multiply the red 2 by the x and then just copy down the blue - 2 without multiplying the red 2 by it to get - 4. Edwin AnlytcPhil@aol.com```
 Quadratic_Equations/42842: I am working on factoring trinomials. The thing is I am so confused with some problems. 2x² + x - 6 can you help me????1 solutions Answer 27904 by AnlytcPhil(1277)   on 2006-06-20 10:36:55 (Show Source): You can put this solution on YOUR website!``` I am working on factoring trinomials. The thing is I am so confused with some problems. 2x² + x - 6 can you help me???? ------------------------------------------------- To factor Ax² ± Bx ± C when there is no common factor of |A|, |B|, and |C| and A is positive. 1. Multiply |A| by |C|, getting AC 2. If the last sign is +, think of two positive integers which have product AC and which have SUM |B| If the last sign is -, think of two positive integers which have product AC and which have DIFFERENCE |B| 3. Rewrite the middle term ±Bx using those two integers found in the preceding step, attaching appropriate signs. 4. Factor by grouping. In your problem, 2x² + 1x - 6, I placed the 1 coefficient beside the middle term for emphasis. 1. Multiply |A| by |C|, getting AC Multiply |2| by |-6| or 2×6 getting 12 2. The last sign is -, so think of two positive integers which have product AC and which have DIFFERENCE |B| So we think of two positive integers which have product 12 and difference of 1. These are 4 and 3, because 4×3=12 and 4-3 = 1 3. Rewrite the middle term ±Bx using those two integers found in the preceding step, attaching appropriate \ signs. We rewrite +1x as +4x - 3x. So now we have 2x² + 4x - 3x - 6 4. Factor by grouping. 2x² + 4x - 3x - 6 Factor 2x out of the first two terms 2x(x + 2) - 3x - 6 Factor -3 out of the last two terms. Notice that you factor out a negative when the next to the last term is preceded by a minus sign. Also when factoring out a nagative, the sign of the last term changes: 2x(x + 2) - 3(x + 2) Notice the common factor (x + 2) which I will color red for emphasis: 2x(x + 2) - 3(x + 2) We factor out the common red factor and leave the black factors inside parentheses: (x + 2)(2x - 3) (x + 2)(2x - 3) -------------------------- Here is another example: Factor 6x² - 19x + 10. 1. Multiply |A| by |C|, getting AC Multiply |6| by |10| or 6×10 getting 60 2. The last sign is +, so think of two positive integers which have product AC and which have SUM |B| So we think of two positive integers which have product 60 and sum of 19. These are 15 and 4, because 15×4=60 and 15+4 = 19 3. Rewrite the middle term ±Bx using those two integers found in the preceding step, attaching appropriate signs. We rewrite -19x as -15x - 4x. So now we have 6x² - 15x - 4x + 10 4. Factor by grouping. 6x² - 15x - 4x + 10 Factor 3x out of the first two terms 3x(2x - 5) - 4x + 10 Factor -2 out of the last two terms. Notice that you factor out a negative when the next to the last term is preceded by a minus sign. Also when factoring out a nagative, the sign of the last term changes: 3x(2x - 5) - 2(2x - 5) Notice the common factor (2x - 5) which I will color red for emphasis: 3x(2x - 5) - 2(2x - 5) We factor out the common red factor and leave the black factors inside parentheses: (2x - 5)(3x - 2) (2x - 5)(3x - 2) Edwin```
 Linear_Algebra/42839: what can be said of points with equal abscissa?with equal ordinates? 1 solutions Answer 27899 by AnlytcPhil(1277)   on 2006-06-20 10:07:30 (Show Source): You can put this solution on YOUR website!``` what can be said of points with equal abscissa? "Abscissa" is a fancy name for the 1st coordinate, or "x-coordinate" Let's plot some and see. Below I have plotted (3,7), (3,3), (3,-4) and (3,-8), all having the same abscissa, 3. Notice that they all lie on the same vertical line. Incidentally the equation of that vertical line is simply this: x = 3 ---------------------------------------------------- with equal ordinates? "Ordinate" is a fancy name for the 2nd coordinate, or "y-coordinate" Let's plot some and see. Below I have plotted (-6,-4), (3,-4), (4,-4) and (7,-4), all having the same ordinate, -4. Notice that they all lie on the same horizontal line. Incidentally the equation of that horizontal line is simply this: y = -4 Edwin```
 Trigonometry-basics/42653: Find the exact value of sin 2θ if cosθ = -(√5)/3 and 180° < θ < 270°. Thank You 1 solutions Answer 27763 by AnlytcPhil(1277)   on 2006-06-18 18:17:42 (Show Source): You can put this solution on YOUR website!```Find the exact value of sin 2θ if cosθ = -(√5)/3 and 180° < θ < 270°. Thank You We need to use the formula sin(2θ) = 2sinθ·cosθ We know cosθ but not sinθ. We have to find sinθ. We are given 180° < θ < 270° so we know θ is in quadrant III. So we draw the picture of θ in the third quadrant _ | -√5 | ------------- | /| | /3| |/ | Since the cosine is x/r or adjacent/hypotenuse, we _ _ put the numerator of -√5/3, namely -√5 the on the x-side (or adjacent side) and 3 on the r, the radius vector (or hypotenuse). We use the Pythagorean theorem to find the y-side (or the opposite side). _______ ____________ ___ _ y = ±√r² - x² = ±√(3)² - (-√5)² = ±√9-5 = ±√4 = ±2 We know to take the negative sign since y goes down from the x-axis, so we have _ | -√5 | ------------- | /| -2| /3| |/ | Now we know sinθ = -2/3 So sin(2θ) = 2sinθ·cosθ _ sin(2θ) = 2(-2/3)·(-√5/3) _ sin(2θ) = 2(-2/3)·(-√5/3) _ sin(2θ) = 4√5/9 Edwin AnlytcPhil@aol.com```
 Trigonometry-basics/42656: Find the exact value of sin θ/2 if cos θ = 2/3 and 270° < θ <360°. Thank You 1 solutions Answer 27755 by AnlytcPhil(1277)   on 2006-06-18 17:20:35 (Show Source): You can put this solution on YOUR website!```Find the exact value of sin θ/2 if cos θ = 2/3 and 270° < θ < 360°. Use the formula __________ sin(θ/2) = ±Ö(1-cosθ)/2 Where the sign of the sine is taken according to which quadrant x is in. It is unfortunate that the words "sine" and "sign" are homophones, because it sounds funny and is often confusing to speak of "the sign of the sine". Some teachers use "positiveness or negativeness of the sine" instead of "sign of the sine". Anyway, substitute 2/3 for cosθ in the formula: __________ sin(θ/2) = ±Ö(1-2/3)/2 Change the last "/" to a "÷" _______ sin(θ/2) = ±Ö(1/3)÷2 Change the 2 to 2/1 ___________ sin(θ/2) = ±Ö(1/3)÷(2/1) Invert and multiply to perform the divion under the radical: ___________ sin(θ/2) = ±Ö(1/3)×(1/2) ___ sin(θ/2) = ±Ö1/6 Now we have to determine whether to use the + or the - . We are givem: 270° < θ < 360° so we whii divide all three sides of that by 2 (270/2)° < θ/2 < (360/2)° 135° < θ/2 < 180° This tells us that θ/2 is in quadrant II. The sine is positive in the second quadrant Therefore we know that sin(θ/2) is positive, So ___ sin(θ/2) = Ö1/6 Your teacher may want you to rationalize the denominator and get _ sin(θ/2) = Ö6/6 Edwin```
 Linear_Algebra/42604: what do points on the line parallel to the y-axis in 3 units to the left of it have in common?with a summary. 1 solutions Answer 27719 by AnlytcPhil(1277)   on 2006-06-18 06:45:30 (Show Source): You can put this solution on YOUR website!``` what do points on the line parallel to the y-axis in 3 units to the left of it have in common?with a summary. Draw the line. Then mark some point arbitrarily and compare their coordinates to see what they have in common. I have marked some points arbitrarily on that line. The top one is (-3,7), the next one down is (-3,3), the bottom one is (-3,-8). Notice that they all have the same x-coodinate, namely -3. That's the answer. All the points on that line have in common the fact that their x-coordinates are -3. Edwin AnlytcPhil@aol.com```
 Complex_Numbers/42605: This question is from textbook I am lost on this problem... radical -50 times radical -21 solutions Answer 27718 by AnlytcPhil(1277)   on 2006-06-18 06:17:17 (Show Source): You can put this solution on YOUR website!```I am lost on this problem... radical -50 times radical -2 ___ __ Ö-50 Ö-2 Don't fall for the trap of always multiplying under radicals first. Don't do that! Negative numbers under square roots must be taken out of the radical before multiplying under them: ___ __ __ _ __ _ ___ Ö-50 Ö-2 = (iÖ50)(iÖ2) = i² Ö50 Ö2 = (-1)Ö100 = (-1)(10) = -10 If you multiply under the radical before taking negative signs out of square root radicals as i's on the outside, you will get the wrong answer. Edwin AnlytcPhil@aol.com```
 expressions/42377: In which year did the greatest percent increase in filings occur? 1 solutions Answer 27556 by AnlytcPhil(1277)   on 2006-06-16 12:21:53 (Show Source): You can put this solution on YOUR website!```In which year did the greatest percent increase in filings occur? Percent increase is figured by the formula (New value minus old value)÷(old value)×100% In 1997 the number of bankruptcies was 14 thousand In 1998 the number of bankruptcies was 16 thousand, which was an increase over the 14 thousand in 1997 of 16-14 or 2 thousand. The percent increase was 2÷14×100% = 14.3%, approximately In 1999 the number of bankruptcies was 21 thousand, which was an increase over the 16 thousand in 1998 of 21-16 or 5 thousand. The percent increase was 5÷16×100% = 31.3%, approximately In 2000 the number of bankruptcies was 24 thousand, which was an increase over the 21 thousand in 1990 of 24-21 or 3 thousand. The percent increase was 3÷21×100% = 14.3%, approximately In 2001 the number of bankruptcies was 30 thousand, which was an increase over the 24 thousand in 2000 of 30-24 or 6 thousand. The percent increase was 6÷24×100% = 25%, exactly. So the greatest percent increase in filings was in 1999 with 31.3% approximate percentage increase over the 16 thousand filed the year before in 1998. Edwin```
 Miscellaneous_Word_Problems/42397: Simplify: (3 1/2) ÷ (2 3/5) × (1 1/4) ÷ (7/13) (show your work) THank you for the help1 solutions Answer 27552 by AnlytcPhil(1277)   on 2006-06-16 11:54:52 (Show Source): You can put this solution on YOUR website!```Simplify: (3 1/2) ÷ (2 3/5) × (1 1/4) ÷ (7/13) (show your work) (3 1/2) ÷ (2 3/5) × (1 1/4) ÷ (7/13) Change the mixed fractions to improper fractions: (7/2) ÷ (13/5) × (5/4) ÷ (7/13) Invert the fractions that are divided and change the division to a multiplication (7/2) × (5/13) × (5/4) × (13/7) or 7 5 5 13 ——— × ———— × ——— × ———— 2 13 4 7 Cancel the 7 numerator in the first fraction and the 7 denominator in the last fraction: 1 7 5 5 13 ——— × ———— × ——— × ———— 2 13 4 7 1 Cancel the 13 denominator in the second fraction and the 13 numerator in the last fraction: 1 1 7 5 5 13 ——— × ———— × ——— × ———— 2 13 4 7 1 1 So you have 1 5 5 1 ——— × ——— × ——— × ——— 2 1 4 1 Multiply the numerators 1×5×5×1 = 25 Multiply the denominators 2×1×4×1 = 8 You get 25 ———— 8 Finally we can change that to a mixed number 3 1/8 Edwin AnlytcPhil@aol.com```
 Trigonometry-basics/42390: Write the equation sin y=x in the form of an inverse function. Thank you1 solutions Answer 27547 by AnlytcPhil(1277)   on 2006-06-16 11:31:22 (Show Source): You can put this solution on YOUR website!```Write the equation sin(y) = x in the form of an inverse function. Thank you The equation in words would be "The sine of y is x" That sentence can be written this way: "y is what we must take the sine of to get x" The mathematical symbol for the words "what we must take the sine of to get" is sin-1 That is not really a "negative one exponent", it's only written that way because a negative one exponent is what inverts a fraction, and that reminds us that we are writing another kind of inverse somewhat analogous to that, but far from meaning that. So from the beginning: sin(y) = x translates as "The sine of y is x" which translates as "y is what we have to take the sine of to get x" which translates as y = sin-1(x) Edwin AnlytcPhil@aol.com```
 Miscellaneous_Word_Problems/42386: which number is a factor of 20? A) 1 B) 12 C) 40 D) 601 solutions Answer 27530 by AnlytcPhil(1277)   on 2006-06-16 11:11:43 (Show Source): You can put this solution on YOUR website!```which number is a factor of 20? A) 1 B) 12 C) 40 D) 60 You must learn that the words "IS A FACTOR OF" are exactly equivalent to the words "WILL DIVIDE EVENLY INTO WITHOUT LEAVING A REMAINDER (OR A REMAINDER OF 0)." So pretend the problem were stated this way: which number WILL DIVIDE EVENLY INTO 20 WITHOUT LEAVING A REMAINDER (OTHER THAN 0)? A) 1 B) 12 C) 40 D) 60 Let's see. We'll start with the last one. Try (D). WILL 60 DIVIDE EVENLY INTO 20 WITHOUT LEAVING A REMAINDER OTHER THAN 0? Let's see: 0 60)20 0 20 No that leaves a remaider of 20 Try (C). WILL 40 DIVIDE EVENLY INTO 20 WITHOUT LEAVING A REMAINDER? Let's see: 0 40)20 0 20 No that also leaves a remaider of 20 Try (B). WILL 12 DIVIDE EVENLY INTO 20 WITHOUT LEAVING A REMAINDER? Let's see: 1 12)20 12 8 No that leaves a remaider of 8 Try (A). WILL 1 DIVIDE EVENLY INTO 20 WITHOUT LEAVING A REMAINDER OTHER THAN 0? Let's see: 20 1)20 2 0 0 0 YUP, that only leaves a remainder of 0, so that's the answer. Edwin AnlytcPhil@aol.com```