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# Recent problems solved by 'jsmallt9'

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 Quadratic_Equations/709189: Find the values of if the Quad. eqn. has 2 real and equal roots. Hence, find the corresponding root of the equation based on each value of that you have found.1 solutions Answer 436570 by jsmallt9(3296)   on 2013-02-02 09:09:25 (Show Source): You can put this solution on YOUR website! First let's put the equation into standard form. We'll start by simplifying the left side: Then we'll gather all the terms on one side (so the other side is zero): Next we will gather and group the terms, the x terms and the "other terms": Factoring out from the first group and the x from the second group we get: To make this look more like standard form I will use the Commutative Property to switch the order of the factors: We now have standard form with... a = m-1 b = m+1 c = m-1 We will get equal roots if (the discriminant) = 0. Replacing the a, b and c we found above into this equation we get: Simplifying we get: Now we solve for m. It will be easier to factor if we make the "a" in this quadratic positive. So we'll start by factoring out -1: Then we can factor more: From the Zero Product Property: -1 = 0 or 3m-1 = 0 or m-3 = 0 The first equation is false and has no solution. The other two equations have solutions: m = 1/3 or m = 3 So if the m in your original equation is either 1/3 or 3 there will be two equal roots to the equation. To find the roots when m = 1/3: Replace the m with 1/3: Now we solve for x. To make things easier I'm going to multiply each side by three to get rid of the fraction: Simplifying... Making one side zero: Factoring: Zero Product Property: 2 = 0 or There is no solution to the first equation. But the second equation has a solution of x = 1. So when m = 1/3 your equation has two equal roots of 1. I'll leave it up to you to figure out the equal roots when m = 3.
 Polynomials-and-rational-expressions/709092: what is the oblique asymptote of f(x)=(7x^3-5x^2-9x+7)/(-6x^2-5x-9)1 solutions Answer 436566 by jsmallt9(3296)   on 2013-02-02 08:38:10 (Show Source): You can put this solution on YOUR website! Find find the oblique asymptote of a rational function you divide the numerator by the denominator so you can rewrite the function in the form: f(x) = quotient + remainder/(f(x)'s denominator) We will not be able to use synthetic division since our function's denominator is quadratic. So we must use long division. [Note: Since -6x^2 does not divide evenly into 7x^3 (and probably other terms too) this is going to get messy with fractions.] ``` (-7/6)x + (-65/36) -6x^2 -5x -9 / 7x^3 -5x^2 -9x + 7 - 7x^3 + (35/6)x^2 + (42/6)x (-65/6)x^2 + (-294/6)x + 7 - (-65/6)x^2 + (325/36)x + 65/4 (-2089/36)x + (-37/4)``` So f(x) in the desired form is: As x get to be very large (positive or negative), the fraction at the end becomes closer and closer to zero. This makes the quotient part the oblique asymptote for large values of x. So the oblique asymptote for f(x) is the line:
 Polynomials-and-rational-expressions/709224: Identify the greatest common factor then factor the expression. 4x^9 + 8x^8 - 12x^7 + 8x^6 1 solutions Answer 436560 by jsmallt9(3296)   on 2013-02-02 07:31:14 (Show Source): You can put this solution on YOUR website!4x^9 + 8x^8 - 12x^7 + 8x^6 If you have trouble finding GCF's, it can be helpful to factor each term "fully". By "fully" I mean:Factor each coefficient into prime numbersRewrite the variables without exponentsLet's see: ```4x^9 = 2 * 2 * x * x * x * x * x * x * x * x * x 8x^8 = 2 * 2 * 2 * x * x * x * x * x * x * x * x 12x^7 = 2 * 2 * 3 * x * x * x * x * x * x * x 8x^6 = 2 * 2 * 2 * x * x * x * x * x * x ``` Note how I used spacing the ensure that each column has the same factor in it. Lining up the factors like this can also be helpful in finding the GCF. In the table above a factor that is common to all four terms will be the factor in any column that has four entries. The GCF will be the product of all the common factors. Looking at the table we can see that there are 2 columns of 2's and 6 columns of x's that have an entry in all four rows. So the GCF is: `GCF = 2 * 2 * x * x * x * x * x * x = 4x^6` P.S. There are other uses for factoring and lining up the factors like this:Figuring out "what's left" of each term if the GCF is factored out. Just look at the GCF line of factors and compare it to a term's line. The factors that are in the term's line but not in the GCF's line of factors will be "what's left" of the term after the GCF is factored out. For example: ```GCF = 2 * 2 * x * x * x * x * x * x 8x^6 = 2 * 2 * 2 * x * x * x * x * x * x```From this we can see that the only factor in the term's line that is not in the GCF line is that 3rd 2. So if the GCF is factored out of 8x^6 there will be a 2 left. Or ```GCF = 2 * 2 * x * x * x * x * x * x 12x^7 = 2 * 2 * 3 * x * x * x * x * x * x * x```Here we can see that the term has factors of a 3 and a 7th x that are not in the GCF list. So if we factor out the GCF from this term we will have 3 * x or just 3x left. Note: If a term is the same as the GCF then there will be just a 1 (one) left if the GCF is factored out of this term. Finding least common multiples (LCM's). The LCM will be the product of all the different columns. For our four terms above the LCM will be: LCM = 2 * 2 * 2 * 3 * x * x * x * x * x * x * x * x * x = 24x^9 Note: Lowest common denominators (LCD's) are simply the LCM of some denominators. So we can use this method to find LCD's, too. Not only that we can figure out each fraction need to be multiplied by to turn the denominator into the LCD. For example, let's say we wanted to add We would need the LCD. Since the denominators are the same as the terms we've already factored we can use the LCM we found above: `LCM = 2 * 2 * 2 * 3 * x * x * x * x * x * x * x * x * x = 12x^9`Then if we compare the factors of the LCM with the factors of each denominator we can see what factors are "missing". For the first denominator: `4x^9 = 2 * 2 * x * x * x * x * x * x * x * x * x`We can see that this term's factors are "missing" the third 2 and a 3. So we would multiply the numerator and denominator of the first fraction by 2*3 or 6. For the third denominator: `12x^7 = 2 * 2 * 3 * x * x * x * x * x * x * x`We can see that this term's factors are missing the third 2 and the 8th and 9th x's. So we would multiply the third fraction's numerator and denominator by 2 * x * x or Note: If a denominator is the same as the LCD then it does not need to change. Don't multiply it by anything.
 Polynomials-and-rational-expressions/709334: ((5x^-2)+(5y^-2))/((8x^-1)+(8y^-1)) Simplify the compound fractional expression. Give your answer in factored form.1 solutions Answer 436559 by jsmallt9(3296)   on 2013-02-02 06:54:04 (Show Source): You can put this solution on YOUR website! This is a compound fraction because of the negative exponents. Let's rewrite the expression with positive exponents so we can see it better: Different methods are taught on how to simplify this expression. One method is toCombine the terms in both the numerator and the denominator of the "big" fractionTurn the division into a multiplication by the reciprocal of the denominatorSimplify But I prefer a different method:Find the lowest common denominator (LCD) of all the "little" fractions.Multiply the numerator and denominator of the "big" fraction by the LCD from step one.The first method requires addition/subtraction of fractions. The second one does not which is a major reason I prefer it. Using the second method... The "little" denominators are , , x and y. The LCD of these 4 is: . Multiplying the numerator and denominator by this LCD: To multiply correctly we need to use the Distributive Property: In each "little" fraction the denominator cancels with all or some part of the LCD leaving: There are no like terms here so we cannot simplify this further. But we might be able to reduce the fraction. To see if the fraction will reduce we must first factor the numerator and denominator: There are no factors in common between the numerator and denominator so the fraction will not reduce. The problem asks that the answer be left in factored form so our answer is:
 Linear-systems/709325: does anyone know how to solve x,y,z equations for x and explain?! 2x + y + z = 0 3x - y + z = 3 7x - 5y - 3z = 15 thanks!1 solutions Answer 436550 by jsmallt9(3296)   on 2013-02-02 02:13:46 (Show Source): You can put this solution on YOUR website!There are many ways to solve a system like this, including:Substitution MethodLinear Combination (aka Elimination) MethodMatrix methods, including:Gaussian EliminationInverse MatricesCramer's Rule (determinants)Since it would not be helpful for me (or any tutor) to use a method you know nothing about, I think it would be best for you to re-post your question and include either:The method you would like to see; orA list of methods you have learned so the tutors can pick one you can recognize.
 Linear-systems/709375: how do I solve by elimination? -0.3x-0.1y=-0.2 2/7x-3/14y=5/141 solutions Answer 436549 by jsmallt9(3296)   on 2013-02-02 02:04:09 (Show Source): You can put this solution on YOUR website! Usually the first thing with the elimination method is get opposites lined up. With the fractions and decimals, however, this will not be simple. So the first thing we will do is eliminate the fractions and decimals. Multiplying both sides of the first equation by 10 will eliminate the decimals. And multiplying both sides of the second equation by 14 will eliminate the fractions: which simplify to: Now that the fractions and decimals are gone we can more easily create lined-up opposites. Multiplying the first equation by -3 will create lined-up opposite y-terms: which simplifies to: After the opposites are lined up the next step is to add the two equations together: With the y terms gone we can solve for x: Now that we have x we can find y. Usually I recommend using one of the original equations at this point. It is the safest way to find the second variable. (Using an equation you created might be a problem if you've made a mistake somewhere along the way.) But with the x being a fraction and the original equations having decimals or fractions with different denominators, I'm going to be lazy and take a chance using of the later equations (without fractions or decimals). I'm going to use: Substituting in our value for x: which simplifies to: Multiplying by 13 to eliminate the fraction: Adding 33: Dividing by -13: So the solution to your system is the point: (, )
 Graphs/709377: suppose that the function g is defined, for all real numbers, as follows. g(x)= -x if x does not equal 3 g(x)=2 if x equals 3 graph the function g.1 solutions Answer 436548 by jsmallt9(3296)   on 2013-02-02 01:33:48 (Show Source): You can put this solution on YOUR website!Pretend for a moment that the function was simply g(x) = -x then I hope you would recognize this as the equation for a line. A line in slope-intercept form. A line with a slope of -1 and a y-intercept of 0. One of the points on this pretend line is (3, -3) Since the real g(x) is -x for all x's except 3, graph the line described above except for the point (3, -3)!. In other words. draw an small open circle (a small ring) at the point (3, -3). (This open circle indicates that this point is not included.) Then draw the rest of the line as normal. When you are finished it should look like the the line y = -x except that it has a "hole" in it at (3, -3). To finish the graph just plot the point (3, 2). (This is a regular point, a dot, not the open circle we used at (3, -3).) Your final graph should be a line with a hole in it at (3, -3) plus the extra point (3, 2) (which is nowhere near the line).
 Trigonometry-basics/709264: Verify that tan theta/ sec theta= sin theta1 solutions Answer 436487 by jsmallt9(3296)   on 2013-02-01 15:44:26 (Show Source): You can put this solution on YOUR website!When you are not sure how to prove a Trig identity (like this), it can be helpful to convert any tan's, cot's, sec's and/or csc's into equivalent expressions in terms of of sin and/or cos. With tan = sin/cos and sec = 1/cos your equation becomes: We can simplify the fractions within a fraction on the left side bu multiplying the numerator and denominator of the "big" fraction by : The cos's cancel: leaving: which simplifies to: And we're finished!
 Permutations/701774: ten cards are numbered 1 through 10. three cards are selected. find the probability that all three cards are less than 5.1 solutions Answer 436472 by jsmallt9(3296)   on 2013-02-01 11:59:07 (Show Source): You can put this solution on YOUR website!I think the other tutor did not notice that three cards are to be drawn. His/her solution is only correct for drawing a single card. To find the probability of all three cards being less than 5:Find the individual probabilities of each card being less than 5; thenMultiply these probabilities.The probability of the first card being less than 5 is, as the other tutor said, 4/10. The probability of the second card being less than 5 is 3/9 since there are only three cards left that are less than 4 and there are only 9 cards altogether after the first card. The probability of the third card being less than 5 is 2/8 since there are only two cards left that are less than 4 and there are only 8 cards altogether after the first two cards. The probability of all three cards being less than 5 is the product of the three individual probabilities: which simplifies down to:
 Volume/709234: how many bricks,each of dimensions 20cm*12cm*9cm,will be required to build a wall 12m long 72cm thick 4m hight.1 solutions Answer 436469 by jsmallt9(3296)   on 2013-02-01 11:39:15 (Show Source): You can put this solution on YOUR website!The solution from the other tutor is correct. But it is only correct because the each side of the wall can be divided evenly by a corresponding side of a brick. So a full, proper solution will first determine this divisibility. The 20cm side of the brick will divide evenly into the 4m side of the wall: 4m = 400cm and 400/20 = 20 The 12cm side of the brick will divide evenly into the 12m side of the wall: 12m = 1200cm and 1200/12 = 100 The 9cm side of the brick will divide evenly into the 72cm side of the wall: 72/9 = 8 This makes the number of bricks needed: 20*100*8 = 16000 (which matches the answer from the other tutor). P.S. When checking for the divisibility, try all possible combinations. In this problem, the 20cm side of the brick divides into both the 12m and 4m sides of the wall. The 12cm side of the brick divides into both the 72cm and 12m sides of the wall. But the 9cm side of the brick only divides evenly into the 72cm side of the wall. This leaves the 12m for the 12cm and then the 20cm into the 4m. P.P.S. If you cannot find this divisibility then it will not be possible to build the wall unless you break off parts of the bricks. This is a more difficult problem to solve and the solution from the other tutor, to divide the volumes, will not work correctly.
 Polynomials-and-rational-expressions/709227: Using complete sentences, describe how you would find all possible rational zeros of the polynomial function f(x) = 9x4 – 17x3 + 2x2 – 3x + 33.1 solutions Answer 436464 by jsmallt9(3296)   on 2013-02-01 11:17:09 (Show Source): You can put this solution on YOUR website!When a polynomial (this is not a rational function by the way) is written in standard form with the terms listed from highest degree to lowest (like yours), the possible rational roots are all the ratios, positive and negative, that can be formed using a factor of the constant term (at the end) in the numerator over a factor of the leading coefficient in the denominator. Your constant term is 33. Its factors are 1, 3, 11 and 33. Your leading coefficient is 9. Its factors are 1, 3 and 9. The possible rational roots of this polynomial are all the possible ratios, positive and negative, we can form using a factor of 33 (1, 3, 11 or 33) on top over a factor of 9 (1, 3 or 9): +1/1, +3/1, +11/1, +33/1, +1/3, +3/3, +11/3, +33/3, +1/9, +3/9, +11/9, +33/9 which reduce to: +1, +3, +11, +33, +1/3, +1, +11/3, +11, +1/9, +1/3, +11/9, +11/3 Removing the duplicates: +1, +3, +11, +33, +1/3, +11/3, +1/9, +11/9 So there are 16 possible rational roots to f(x).
 Polynomials-and-rational-expressions/709135: how do you find all of the real zeroes of 8x^3+1251 solutions Answer 436462 by jsmallt9(3296)   on 2013-02-01 10:53:50 (Show Source): You can put this solution on YOUR website! To find the zeroes of an expression of degree 3 or more (like this one) we need to factor it. When factoring, check the greatest common factor (GCF) first. The GCF here is 1 (which we rarely factor out). After the GCF, we look to use any and all of the other factoring techniques. One of these techniques is factoring by the use of factoring patterns. And one of these patterns (the only one in fact) that has two terms with a "+" between them is: As the pattern shows, we need a sum of two cubes to use this pattern. Is a sum of cubes? It's definitely a sum. Are and perfect cubes? With a little investigation we should find that and . So we can use the pattern with the "a" being "2x" and the "b" being "5". So , according to the patterns factors into: (Note: My use of parentheses may seem excessive. But it is actually a good idea to use parentheses like this when making substitutions like we have done here. It helps us avoid easily-made mistakes.) This simplifies to: Neither factor will factor further no matter which techniques we try. From the Zero Product Property we know that this product will be zero only be zero if one of the factors is zero. And if this product is zero then the expression from which we got these factors, will also be zero. So the x values that make these factors zero will be the zeroes of . Setting each factor to zero and then solving will lead to our solution. For the first factor: Solving this we should get x = -5/2. For the second factor: We must use the Quadratic Formula: simplifying... At this point we can stop. The negative in the square root means that the solutions will be complex numbers. But the problem specifically asks for real solutions. So the only real zero to is -5/2. P.S. In case you are curious about the complex zeroes I will go ahead and finish with the Quadratic Formula. Continuing the simplifying: which is short for: or In standard "a + bi" form these would be: or So there are three zeroes: two complex (above) and the one real we found earlier (-5/2).
 Trigonometry-basics/709118: find the exact value of each expression 20) inverse cos( inverse sin root 2 over 3)1 solutions Answer 436444 by jsmallt9(3296)   on 2013-02-01 09:12:03 (Show Source): You can put this solution on YOUR website!If you posted the actual question you have then it is a bad question for several reasons:What is "root"? Square root? cube root? 4th root? etc.What is inside the radical (or whatever kind of root it is)? Just the 2? or 2/3?The inverse functions take a ratio as input and provide an angle as output. Depending on what "root 2 over 3" means, it may be a valid ratio to be input to the inverse sin. But inverse sin will output an angle which is not valid input to inverse cos.If what you posted is not literally the problem given to you (which is waht I suspect), then please:Be more careful in posting. There is virtually no chance an inverse sin is input to an inverse cos.Don't just say "root". Tell us what kind of root.Use parentheses to tell us what is inside the radical of the root. Use square root(2)/3 (or just sqrt(2)/3) for and use square root(2/3) (or just sqrt(2/3)) for
 Trigonometry-basics/709201: From the top of a 130m high building, the angle of depression of Jenny and Johnny on the ground were found to be 34 degrees 16 minutes and 54 degrees 12 minutes, respectively. Find the distance between them. Can you also please help me with illustrating this? I have a hard time illustrating triangles. Thanks. 1 solutions Answer 436441 by jsmallt9(3296)   on 2013-02-01 08:43:00 (Show Source): You can put this solution on YOUR website!This problem is just like your plane and ships problem. (You can even reuse the diagram as long as you change the angles and the heights!) So you can solve it the same way as I showed you in that problem. The only substantial difference between this problem and the other one is that the angles are not whole numbers. To handle angles measured in degrees and minutes you must convert them into a decimal number of degrees. Since there are 60 minutes in a degree, 34 degrees and 16 minutes is the same as degrees. Now we convert the fraction into a decimal by dividing 16/60. So 34 degrees 16 minutes, as a decimal number of degrees, is: 34.266666... Round this off as you see fit. I'll leave it up to you to convert 54 degrees 12 minutes to a decimal number of degrees. Then as you solve the problem you will be using these decimal numbers in your tan's.