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Quadratic-relations-and-conic-sections/452786: Can you help me write this in standard form? y=x^2+2x+2 1 solutions Answer 311180 by jim_thompson5910(28696)   on 2011-05-22 19:31:15 (Show Source): You can put this solution on YOUR website! To write in standard form, we must complete the square for Start with the given expression. Take half of the coefficient to get . In other words, . Now square to get . In other words, Now add and subtract . Make sure to place this after the "x" term. Notice how . So the expression is not changed. Group the first three terms. Factor to get . Combine like terms. So after completing the square, transforms to . So . So is equivalent to . which is now in standard form.

expressions/452716: Factor x^2+6x+9 1 solutions Answer 311179 by jim_thompson5910(28696)   on 2011-05-22 19:29:04 (Show Source): You can put this solution on YOUR website! Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is . Now multiply the first coefficient by the last term to get . Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ? To find these two numbers, we need to list all of the factors of (the previous product). Factors of : 1,3,9 -1,-3,-9 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*9 = 9 3*3 = 9 (-1)*(-9) = 9 (-3)*(-3) = 9 Now let's add up each pair of factors to see if one pair adds to the middle coefficient : First Number Second Number Sum 1 9 1+9=10 3 3 3+3=6 -1 -9 -1+(-9)=-10 -3 -3 -3+(-3)=-6 From the table, we can see that the two numbers and add to (the middle coefficient). So the two numbers and both multiply to and add to Now replace the middle term with . Remember, and add to . So this shows us that . Replace the second term with . Group the terms into two pairs. Factor out the GCF from the first group. Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis. Combine like terms. Or factor out the common term Condense the terms. =============================================================== Answer: So factors to . In other words, . Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Polynomials-and-rational-expressions/452775: I don't understand to factor this out (4h-16)/(h^2-16) i put it in the calc but i dont understand how it factors out h-4 . I factor out 4 from the top and ^2 from the bottom but i get the wrong answer please help thanks, wildcworkz 1 solutions Answer 311178 by jim_thompson5910(28696)   on 2011-05-22 19:27:59 (Show Source): You can put this solution on YOUR website! So simplifies to In other words,

Triangles/452746: For a triangle with a perimeter of 12 and an area of 7 what are the lengths of the triangles legs? Is there an equation that relates area and perimeter to give you leg lengths? Or something similar? Some things I’ve tried: 1: Finding “a” “b” and “c” directly by converting a set of equations into a matrix in row echelon form. The issue is I can’t find a set of linear equations that relate a, b, and c to the triangle. 2: Listing possible solutuions for b and h from the area formula that are also factors of 14. 14 because . This is impossible because there are an infinite number of possibilities. So I went for positive integers [(b:1,2,7,14)(h:1,2,7,14)]. I didn't continue with this train of thought because something else occured to me. 3: When a triangle is split into 2 triangles and the line that splits them is perpedicular to one of the lines it touches then both of the triangles are right triangles. The pythagorian therum works for right triangles. Giving me another equation to work with. I couldn't figure out how to use it though. 4:So I square 14 giving me 196. Factoring 196 into primes gave me (2,2,7,7). I made a chart of the ways these numbers could be combined so when they were multiplied i got 14. This gave me , , and I came to the same problem again as i had with the pythagorean therum; i didn't know how to use the new information. 5: So i went back to the visually drawn triangles and labeled the sides of the original triangle a, b, and c. Then I labeled one of the split triangles sides (the numbers to the right of the letters are subscripts) a1, b1, and c1. The other right triangle was labeled a2, b2, and c2. With the sides labeled i tried to substitute various variables in for each other to see what i could come up with. For example the 2 new triangles both have a side that is equal to (1/2)c from the original triangle so i replace the corresponding variable on the smaller triangle with (1/2)c. I tried many more substitutions like that but none of them yeilded anything I thought I could use. I am now taking a break from it. And it occurred to me that if there were an equation relating area with perimeter that would be very usefull. Also if there are any other equations that you think might be useful please send those to me as well. One last thing, please put the things you think might help at the top of the email, and if you know how to solve this kind of problem and include the answer or the "how to" please write something along the lines of "spoiler alert" and skip some lines so it won't be visible. Sorry if i come across as a bit pushy but i really want to figure this one out and then see if i was right. 1 solutions Answer 311169 by jim_thompson5910(28696)   on 2011-05-22 19:00:12 (Show Source): You can put this solution on YOUR website! Hint: The perimeter of a triangle with sides 'a', 'b', and 'c' is (ie add up the sides to get the perimeter). But you probably already knew that. What you probably don't know, or maybe have seen and forgot, is that the area of a triangle with sides 'a', 'b', and 'c' is where (this is the semiperimeter, which is half the perimeter) Let me know if this is enough to help get you started. Spoiler Alert: I don't think there's a solution because solving for a,b, and c gives equations which can be graphed. The graphs can then show where the positive solutions lie. It turns out that not all the solutions are positive, which means that a triangle can't be constructed with these given properties.

Volume/452707: what is the volume of a circle with a diameter of 10 and a height of 16 1 solutions Answer 311143 by jim_thompson5910(28696)   on 2011-05-22 17:13:38 (Show Source): You can put this solution on YOUR website! This sounds like a cylinder. Since the diameter is 10, the radius is Start with the volume of a cylinder formula. Plug in and . Replace with (note: Use more digits of to get better accuracy). Square to get . Multiply , and to get (this value is approximate). Round to the nearest thousandth.

Quadratic_Equations/451301: Hi! I was trying to solve this problem for quadratic equation: but in order to simplify it I get stuck on the part during the equation that looks like this (the + sign in the equation represents the + or - one in the actual equation: . I know that the square root of a negative number is (square root of the number as if it were positive) x i(the square root of - 1). I also know that there will be 2 solutions to this equation. But what do I do after if I need to simplify it as much as possible? Thanks for any help you may give me! 1 solutions Answer 310352 by jim_thompson5910(28696)   on 2011-05-18 17:07:04 (Show Source): You can put this solution on YOUR website! Start with the given equation. Notice that the quadratic is in the form of where , , and Let's use the quadratic formula to solve for "x": Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Subtract from to get (this is where you made your mistake). Multiply and to get . Take the square root of to get . or Break up the expression. or Break up the fraction for each case. or Reduce. So the solutions are or For more help with the quadratic formula, see this solver.

Numbers_Word_Problems/451291: I am a 3-digit number less than 300. I am divisible by 2 and 5, but not 3. The sum of my digits is 7. What number am I? 1 solutions Answer 310351 by jim_thompson5910(28696)   on 2011-05-18 17:00:52 (Show Source): You can put this solution on YOUR website! Since the number is "divisible by 2 and 5", this means that the number is divisible by 10 (since 2*5 = 10). So the number ends with a 0. So the possible numbers are: 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290 Now add up the digits of each possible number. The digits that add to 7 will give us the number. It turns out that 160 and 250 fit this description since 1+6+0=7 and 2+5+0=7 So the numbers are 160 and 250.

Linear-equations/451296: Write an equation of the line the passes through (-5,8) and (-3,-8) 1 solutions Answer 310350 by jim_thompson5910(28696)   on 2011-05-18 16:56:16 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Note: is the first point . So this means that and . Also, is the second point . So this means that and . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Reduce So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Rewrite as Distribute Multiply Add 8 to both sides. Combine like terms. Simplify So the equation that goes through the points and is Notice how the graph of goes through the points and . So this visually verifies our answer. Graph of through the points and

Linear-equations/451297: Write an equation of the line the passes through (-5,8) and (-3,-8) 1 solutions Answer 310349 by jim_thompson5910(28696)   on 2011-05-18 16:56:04 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Note: is the first point . So this means that and . Also, is the second point . So this means that and . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Reduce So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Rewrite as Distribute Multiply Add 8 to both sides. Combine like terms. Simplify So the equation that goes through the points and is Notice how the graph of goes through the points and . So this visually verifies our answer. Graph of through the points and

Surface-area/451294: what is the surface area of a sphere with a diameter of 18 centimeters? express the answer in square meters. Round to the nearest hundreth. 1 solutions Answer 310348 by jim_thompson5910(28696)   on 2011-05-18 16:55:16 (Show Source): You can put this solution on YOUR website! If the diameter is 18 cm, then the radius is cm. Start with the surface area of a sphere formula. Plug in . Replace with (note: Use more digits of to get better accuracy). Square to get . Multiply and to get . Multiply and to get . Round to the nearest hundredth. So the surface area of a sphere with a diameter of cm is approximately square cm. I'll let you do the conversion to square meters.

test/451264: The graph of y=ax^2+bx+c intersects the x-axis at two points. Which statement is true about the seros of the function? 1 solutions Answer 310344 by jim_thompson5910(28696)   on 2011-05-18 16:07:34 (Show Source): You can put this solution on YOUR website! If the graph of y=ax^2+bx+c intersects the x-axis at two points, then there are two distinct real zeros. This is because the x-intercepts correspond to the zeros.

Volume/451259: a sphere has a diameter that is 7.36 inches long. finf the volume to the nearest tenth 1 solutions Answer 310343 by jim_thompson5910(28696)   on 2011-05-18 15:55:47 (Show Source): You can put this solution on YOUR website! Since the diameter is 7.36 inches, the radius is inches. Start with the volume of a sphere formula. Plug in . Replace with (note: Use more digits of to get better accuracy). Cube to get . Multiply and to get (this figure is approximate). Multiply and to get (again, this figure is approximate). Round to the nearest tenth. So the volume is approximately 208.8 cubic inches.

Expressions-with-variables/451255: simplify the expression 28+3(m-4)-m 1 solutions Answer 310341 by jim_thompson5910(28696)   on 2011-05-18 15:45:46 (Show Source): You can put this solution on YOUR website! So

Expressions-with-variables/451251: simplify the expression 5x-12+y+x+15-6y 1 solutions Answer 310333 by jim_thompson5910(28696)   on 2011-05-18 15:32:38 (Show Source): You can put this solution on YOUR website! So

Expressions-with-variables/451248: simplify the expression: -(9+4c)-4(c+2)+20 1 solutions Answer 310331 by jim_thompson5910(28696)   on 2011-05-18 15:23:03 (Show Source): You can put this solution on YOUR website! So

Matrices-and-determiminant/451246: what is the domain of f(x)=5/(x-2) 1 solutions Answer 310328 by jim_thompson5910(28696)   on 2011-05-18 15:21:36 (Show Source): You can put this solution on YOUR website! You can't divide by zero. So which means that . So the domain is the set of all real numbers but .

Volume/451242: If I know the volume of a cylinder and the volume of each ball in the cylinder can I find out how many balls the cylinder will hold? 1 solutions Answer 310326 by jim_thompson5910(28696)   on 2011-05-18 15:20:48 (Show Source): You can put this solution on YOUR website! Divide the volume of the cylinder by the volume of a single ball to find the approximate number of balls you can fit in there. Remember to round down to the nearest integer. For example, if a cylinder has a volume of 30 cubic inches and one ball has a volume of 7 cubic inches. Then which rounds down to 4. So the cylinder can hold at most 4 balls. Note: this is of course assuming that the shapes match (ie the radius and height of the cylinder >= the radius of a single ball)

Complex_Numbers/451157: Divide. (4+2i)/(5-6i) Write your answer as a complex number in standard form. 1 solutions Answer 310323 by jim_thompson5910(28696)   on 2011-05-18 15:17:10 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the fraction by . Combine the fractions. FOIL the numerator. FOIL the denominator. Multiply. Replace with -1. Multiply. Combine like terms. Break up the fraction. Reduce. So . So the expression is now in standard form where and

sets-and-operations/451236: I am having some problems understanding this .. this is all new could i get someone to help....thank you Given A = {1, 2, 3, 4}, B = {4, 5, 6,}, and C = {2, 6, 7}. Evaluate each set a) A ∩ B b) A U C c) B U C d) (A U B) ∩ C e) A U (B U C) f) (A ∩ B) ∩ C g) (A ∩ B) U C 1 solutions Answer 310318 by jim_thompson5910(28696)   on 2011-05-18 15:12:32 (Show Source): You can put this solution on YOUR website! I'll do the first three to get you started a) To form the intersection of the two sets A={1,2,3,4} and B={4,5,6}, simply write down all of the elements that are in BOTH sets at the same time. So let's place the first set right above the second set (for an easy visual comparison): {1,2,3,4} {4,5,6} Now highlight the elements that are in BOTH sets: {1,2,3,4} {4,5,6} Since the highlighted terms are 4, this means that the intersection of the two sets is {4} So if and , then the intersection of the two sets is: ========================================================================= b) To form the union of the two sets A={1,2,3,4} and C={2,6,7}, simply make a new set that contains EVERY element in both sets like so {1,2,3,4,2,6,7}. Take note how the red elements are from the first set while the green elements are from the second set. Now remove any duplicate elements to get the set {1,2,3,4,6,7} So if and , then the union of the two sets is: ========================================================================= c) To form the union of the two sets B={4,5,6} and C={2,6,7}, simply make a new set that contains EVERY element in both sets like so {4,5,6,2,6,7}. Take note how the red elements are from the first set while the green elements are from the second set. Now remove any duplicate elements to get the set {4,5,6,2,7}. Now if you want, you can sort the elements from least to greatest to get {2,4,5,6,7} So if and , then the union of the two sets is: For more help with set union and intersection, check out this solver

Probability-and-statistics/451226: Given U = {22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, A = {22, 24, 26, 28, 30}, and B = {23, 24, 25, 28, 29}. Find A′U B′. finding A’, then B’ then find their union. 1 solutions Answer 310312 by jim_thompson5910(28696)   on 2011-05-18 14:57:46 (Show Source): You can put this solution on YOUR website! A = {22, 24, 26, 28, 30} A' = {23, 25, 27, 29, 31, 32} .... this is the set of U but excluding everything in A (ie it's everything but set A). So start with U, then delete every element found in A. B = {23, 24, 25, 28, 29} B' = {22, 26, 27, 30, 31, 32} Now union A' and B' to get A' U B' = {22, 23, 25, 26, 27, 29, 30, 31, 32} This is just the combination of the two sets (ie dump all the elements from both sets A' and B' into A' U B' and remove any duplicates).

Subset/450991: Let A and B be subsets of U with n(A)=12, n(B)=25, n(A')=67, and n(A intersect B)= 11 Find n(A U B'). I have created the Venn diagram and with two circles, where A is 12, B is 25 and the intersection of A and B is 11, but i'm not sure where to go from there. A' is given as 67, but I don't see how that provides me what I need to solve. Thanks 1 solutions Answer 310210 by jim_thompson5910(28696)   on 2011-05-18 04:29:09 (Show Source): You can put this solution on YOUR website! Since and , this means that So which means that So there are 79 elements in the universe U. This is a very important piece of information. Now comes the task of creating the Venn diagram. This isn't required, but it really helps to see what's going on. So start with drawing a rectangle. Now draw two overlapping circles within this rectangle. Label the circles as A and B and label the rectangle as U like so: Now because , this means that the number 11 goes in the region between the two circles (since this is where the two "intersect"). So fill in this region: Now, we know that n(A) = 12 and we know that . We want to know how many is in A alone and not in B. So just subtract off 11 from 12 to get 12-11 = 1. So there is 1 element in set A. So fill in the circle A with "1" like so: Do the same for set B to get 25 - 11 = 14. So there are 14 elements that are only in B (and not in A). Fill this in to get Now add up the numbers in the separate regions to get: 1+11+14 = 26 So there are 26 elements that belong in either set A, set B, or both. Since there are 79 elements total, this must mean that there are 79 - 26 = 53 elements that are in neither set. This number gets written outside the circles but inside the rectangle like so: Now our Venn diagram is done. Our task is to now appropriately shade the region that pertains to . We do this by first shading all of set A (the first set listed) like so then we shade , which is everything but B, to get Combine these shadings to get the final product of What we get is that practically everything is shaded except the region that has a 14 in it. Now just add up the numbers that lie in the shaded regions to get: 1+11+53 = 65 So there are 65 things in the set So

expressions/450717: 4sqrt(75) 1 solutions Answer 310023 by jim_thompson5910(28696)   on 2011-05-17 16:42:09 (Show Source): You can put this solution on YOUR website! So

Graphs/450462: I have a question they want me to use the multiplication pronciple to solve this equation -6x<-18 I was thinking that you would divide both sides by -6 and the answer would be 3 not sure if it would be negative or positive or maybe I am going about it all wrong. 1 solutions Answer 309826 by jim_thompson5910(28696)   on 2011-05-16 22:41:37 (Show Source): You can put this solution on YOUR website! If you divide both sides by -6, you'll get (you have to flip the sign when dividing both sides by a negative number).

Equations/450180: 9j+35=4j 1 solutions Answer 309701 by jim_thompson5910(28696)   on 2011-05-16 16:05:13 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is

Volume/450167: What is the approximate volume of the cone? Use 3.14 8cm by 12 cm 1 solutions Answer 309698 by jim_thompson5910(28696)   on 2011-05-16 15:54:36 (Show Source): You can put this solution on YOUR website! I'm assuming the diameter is 8 cm and the height is 12 cm. So if the diameter is 8 cm, then the radius is 4 cm. Start with the volume of a cone formula. Plug in and Replace with (note: Use more digits of to get better accuracy). Square to get Multiply and to get (this figure is approximate). Multiply and to get (again, this is approximate). Multiply and to get So the volume of the cone with radius and height is approximately cubic centimeters.

Graphs/450164: I'm learning about graphing equations, I would like to know step by step on how to solve this problem: x-2y=1. Please & thank you. 1 solutions Answer 309694 by jim_thompson5910(28696)   on 2011-05-16 15:37:40 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract from both sides. Rearrange the terms. Divide both sides by to isolate y. Break up the fraction. Reduce. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 1 and the run is 2. This means that to go from point to point, we can go up 1 and over 2 So starting at , go up 1 unit and to the right 2 units to get to the next point Now draw a line through these points to graph So this is the graph of through the points and

Square-cubic-other-roots/450076: I'm not sure this belongs here but... For the functions f(x)= sqrt 12x and g(x)= sqrt 3x. Evaluate each expression 1. (f+g)(3) 2. (f-g)(x) 3. (fg)(x) 4. (f/g)(x) Thanks, I'm completely in the dark on this one. 1 solutions Answer 309691 by jim_thompson5910(28696)   on 2011-05-16 15:27:10 (Show Source): You can put this solution on YOUR website! 1. ======================================================== 2. =================================================================== 3. Note: x is a nonnegative number. =================================================================== 4.

Trigonometry-basics/449907: Is 162 degrees converted to radians 9pi/10? 1 solutions Answer 309503 by jim_thompson5910(28696)   on 2011-05-16 00:19:39 (Show Source): You can put this solution on YOUR website! To convert from degrees to radians, multiply by So degrees is equal to radians. So you are correct. Good job.

Circles/449736: what is the locus of points that are 10 centimeters from a given point 1 solutions Answer 309351 by jim_thompson5910(28696)   on 2011-05-15 19:06:33 (Show Source): You can put this solution on YOUR website! This is a circle since every point on a circle is the same distance away from the center. In this case, the circle has a radius of 10 cm.

sets-and-operations/449748: Given U = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, A = {16, 18, 20, 22}, and B = {17, 19, 20, 23, 24}. Find A′U B′. Do not skip to the answer. Show your work, finding A’, then B’ then find their union 1 solutions Answer 309349 by jim_thompson5910(28696)   on 2011-05-15 19:05:03 (Show Source): You can put this solution on YOUR website! A = {16, 18, 20, 22} A' = {15, 17, 19, 21, 23, 24, 25} .... this is the set of U but excluding everything in A (ie it's everything but set A). So start with U, then delete every element found in A. B = {17, 19, 20, 23, 24} B' = {15, 16, 18, 21, 22, 25} Now union A' and B' to get A' U B' = {15, 16, 17, 18, 19, 21, 22, 23, 24, 25} This is just the combination of the two sets (ie dump all the elements from both sets A' and B' into A' U B' and remove any duplicates).

Equations/449749: How do I slove and check this equation 4x-8=5x 1 solutions Answer 309346 by jim_thompson5910(28696)   on 2011-05-15 19:00:39 (Show Source): You can put this solution on YOUR website! Start with the given equation. Add to both sides. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is