#$&* course Mth 158 just a note: thanks for letting me know that you have received my test. no apologies necessary on the grade status, as i understand you have many classes and students that require attention, i was just was worried i had done something incorrectly and you had not gotten it. thanks again, Shane. If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
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Given Solution: * * Starting with (1-2x)^(1/3)-1=0 add 1 to both sides to get (1-2x)^(1/3)=1 then raise both sides to the power 3 to get [(1-2x)^(1/3)]^3 = 1^3. Since [(1-2x)^(1/3)]^3 = (1 - 2x) ^( 1/3 * 3) = (1-2x)^1 = 1 - 2x we have 1-2x=1. Adding -1 to both sides we get -2x=0 so that x=0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: * 1.4.28 (was 1.4.18). Explain how you found the real solutions of the equation sqrt(3x+7) + sqrt(x+2) = 1. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this problem we multiply by a degree of 2 to eliminate the sqrts. Once this is done we have 3x+7+x+2=1, now we combine like terms giving 4x+9=1 now we isolate the x variable by subtracting 9 from both sides giving 4x=-8 then we divide both sides by 4 giving x=-2. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Starting with sqrt(3x+7)+sqrt(x+2)=1 we could just square both sides, recalling that (a+b)^2 = a^2 + 2 a b + b^2. This would be valid but instead we will add -sqrt(x+2) to both sides to get a form with a square root on both sides. This choice is arbitrary; it could be done either way. We get sqrt(3x+7)= -sqrt(x+2) + 1 . Now we square both sides to get sqrt(3x+7)^2 =[ -sqrt(x+2) +1]^2. Expanding the right-hand side using (a+b)^2 = a^2 + 2 a b + b^2 with a = -sqrt(x+2) and b = 1: 3x+7= x+2 - 2sqrt(x+2) +1. Note that whatever we do we can't avoid that term -2 sqrt(x+2). Simplifying 3x+7= x+ 3 - 2sqrt(x+2) then adding -(x+3) we have 3x+7-x-3 = -2sqrt(x+2). Squaring both sides we get (2x+4)^2 = (-2sqrt(x+2))^2. Note that when you do this step you square away the - sign. This can result in extraneous solutions. We get 4x^2+16x+16= 4(x+2). Applying the distributive law we have 4x^2+16x+16=4x+8. Adding -4x - 8 to both sides we obtain 4x^2+12x+8=0. Factoring 4 we get 4*((x+1)(x+2)=0 and dividing both sides by 4 we have (x+1)(x+2)=0 Applying the zero principle we end up with (x+1)(x+2)=0 so that our potential solution set is x= {-1, -2}. Both of these solutions need to be checked in the original equation sqrt(3x+7)+sqrt(x+2)=1 As it turns out: the solution -1 gives us sqrt(4) + sqrt(1) = 1 or 2 + 1 = 1, which isn't true, while the solution -2 gives us sqrr(1) + sqrt(0) = 1 or 1 + 0 = 1, which is true. x = -1 is an extraneous solution that was introduced in our squaring step. Thus our only solution is x = -2. ** STUDENT QUESTION I got to the third step but I got confused on what to eliminate or substitute in, looking at the solution, im still a little confused on how it all worked out. U got any suggestions on how to look at it in a better way??? INSTRUCTOR RESPONSE You're pretty much stuck with this technique and this way of looking at the problem. It should be pretty clear to you that (sqrt(x+3))^2 is just x + 3. Squaring the expression [ -sqrt(x+2) +1] is a little more challenging. We could use the distributive law: [ -sqrt(x+2) +1]^2 = [-sqrt(x + 2) + 1 ] * [-sqrt(x + 2) + 1 ] = -sqrt(x+2) * [-sqrt(x + 2) + 1 ] + 1 * [-sqrt(x + 2) + 1 ] = -sqrt(x+2) * (-sqrt(x + 2) ) + (-sqrt(x + 2) * 1 + 1 * (-sqrt(x + 2) ) + 1 * 1 = (x + 2) - sqrt(x + 2) - sqrt(x + 2) + 1 = x+2 - 2sqrt(x+2) +1. Once we get the equation 3x+7= x+2 - 2sqrt(x+2) +1 we see that we still need to 'get to' that x within the square root. To do that we rearrange the equation so that the square root is on one side, all by itself, so we can square it without dragging a lot of other stuff along. So we do a couple of steps and we get 3x+7-x-3 = -2sqrt(x+2). If we square both sides of this equation, we get rid of all the square roots and we get x out where we can deal with it. The details are in the given solution, but we get the equation 4x^2+16x+16= 4(x+2). This equation now has x^2 and x terms, so we know it's a quadratic, and we rearrange and solve it as such. The details are in the given solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):my final answer came out right, seeing as -1 would not work and -2 did, but my method in solving the problem was completely wrong im afraid. Is the method I used ok or was this just luck that it worked out????????? ------------------------------------------------ Self-critique Rating:2
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Given Solution: * * Here we can factor x^(1/4) from both sides: Starting with x^(3/4) - 9 x^(1/4) = 0 we factor as indicated to get x^(1/4) ( x^(1/2) - 9) = 0. Applying the zero principle we get x^(1/4) = 0 or x^(1/2) - 9 = 0 which gives us x = 0 or x^(1/2) = 9. Squaring both sides of x^(1/2) = 9 we get x = 81. So our solution set is {0, 81). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: * 1.4.46 (was 1.4.36). Explain how you found the real solutions of the equation x^6 - 7 x^3 - 8 =0 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Is this problem we have to, for lack of a better word, trick the equation into an equation that is easier to solve. To do this we let another variable represent a portion of x. so we will use b=x^3, when we do this it changes the equation to b^2-7b-8=0. Now we can easily factor this giving us (b+1)(b-8)=0 now we set these equal to zero giving b=-1 and b=8. Now we have to account for the b representing x^3 so x^3=-1 and x^3=8 giving x=-1^(1/3) and x=8^(1/3). Since 8 is a perfect cube it becomes x=2 and the other becomes x=-1 because -1*-1=1 and then 1*-1=-1. So our solutions are x=-1 and x=2. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Let a = x^3. Then a^2 = x^6 and the equation x^6 - 7x^3 - 8=0 becomes a^2 - 7 a - 8 = 0. This factors into (a-8)(a+1) = 0, with solutions a = 8, a = -1. Since a = x^3 the solutions are x^3 = 8 and x^3 = -1. We solve these equations to get x = 8^(1/3) = 2 and x = (-1)^(1/3) = -1. STUDENT QUESTION I am confused as to why you substituted the a. I know how to do this on the calculator by using y = and 2nd graph to get the solution (-1, 2) INSTRUCTOR RESPONSE If you substitute a for x^3, then you end up with a quadratic equation that can be easily factored. If a = x^3, then x^6 = a^2 so the equation becomes a^2 - 7 a - 8 = 0. We factor this and find that a can be either 8 or -1. So x^3 can be either 8 or -1. Thus x can be either 2 or -1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: * 1.4.64 (was 1.4.54). Explain how you found the real solutions of the equation x^2 - 3 x - sqrt(x^2 - 3x) = 2. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this we use the method used in the previous problem, so we substitute a for sqrt(x^2-3x) giving the problem a^2-a=2. Now we subtract 2 from both sides and get a^2-a-2=0, now we can factor giving (a-2)(a+1)=0. Now we set these terms equal to 0 giving a=2 and a=-1. We have to remember that we substituted a for sqrt(x^2-3x) so we bring that back in. So sqrt(x^2-3x)=-1 and sqrt(x^2-3x)=2 and considering we cant have a negative sqrt the first is not an option. So sqrt(x^2-3x)=2 simplifies to x^2-3x=4 when we multiply everything by a degree of 2. Then we subtract 4 from both sides giving x^2-3x-4=0 and now we factor giving (x-4)(x+1)=0 and setting each term equal to 0 we get x=4 and x=-1. confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Let u = sqrt(x^2 - 3x). Then u^2 = x^2 - 3x, and the equation is u^2 - u = 2. Rearrange to get u^2 - u - 2 = 0. Factor to get (u-2)(u+1) = 0. Solutions are u = 2, u = -1. Substituting x^2 - 3x back in for u we get sqrt(x^2 - 3 x) = 2 and sqrt(x^2 - 3 x) = -1. The second is impossible since sqrt can't be negative. The first gives us sqrt(x^2 - 3x) = 2 so x^2 - 3x = 4. Rearranging we have x^2 - 3x - 4 = 0 so that (x-4)(x+1) = 0 and x = 4 or x = -1. STUDENT QUESTION I got stuck on this part u=(-sqrt2+-sqrt10)/2, but after I looked at the solution, it made a little more sense to me, but im not real confident. Got any suggestions on how to approach it in a different way??? INSTRUCTOR RESPONSE Plugging into the quadratic formula we get u=(-sqrt2+-sqrt10)/2, meaning u can take one of the two values u=(-sqrt2+sqrt10)/2 or u=(-sqrt2-sqrt10)/2. These quantities are just plain old numbers, which you could evaluate (up to some roundoff) on your calculator. The first possible value of u is about equal to about .874. The second possible value of u is negative. Now u stands for x^2, so we ignore the negative value of u (this since x^2 can't be negative). So we're left with x^2 = u = .874. So x = +- sqrt(.874), giving us the values of x in the given solution. STUDENT QUESTION I still do not understand using u. I can do it from the 2nd step. Problem: u^2 - u - 2 = 0. Factor: (u-2)(u + 1) = 0. You get u = 2, -1. You will solve x^2 - 3x -4. Factor will be (x - 4)(x + 1) = 0. Solutions are 4, -1. INSTRUCTOR RESPONSE The left-hand side consists of x^2 - 3x and the square root of x^2 - 3x. So instead of x^2 - 3 x - sqrt(x^2 - 3x) we write the left-hand side as u - sqrt(u), which is easier to deal with. We solve for u, then come back and figure out what value(s) of x give us our values of u. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):I arrived at the correct answer but there were so many steps I was afraid I had gone wrong somewhere. ------------------------------------------------ Self-critique Rating:2 ********************************************* Question: * 1.4.92 \ 90 (was 1.4.66). Explain how you found the real solutions of the equation x^4 + sqrt(2) x^2 - 2 = 0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this problem I first substituted a=x^2 making the problem a^2+sqrt(2)a-2=0 then I tried to put this into the quadratic formula which I know as -b(+)(-)sqrt(b)^2-4*a*c/2*a. when I tried this however I ran into problems that did not match your given solution. I think it seems that the a variable is not present in your given solution and a sqrt root that im not sure where it came from. The way I understand to plug in the variables would be -sqrt(2)(+)(-)sqrt(sqrt(2)^2-4*u^2*-2/2*u^2. Perhaps you can show me what I have done wrong.
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Given Solution: * * Starting with x^4+ sqrt(2)x^2-2=0 we let u=x^2 so that u^2 = x^4 giving us the equation u^2 + sqrt(2)u-2=0 Using the quadratic formula u=(-sqrt2 +- sqrt(2-(-8))/2 so u=(-sqrt2+-sqrt10)/2 Note that u = (-sqrt(2) - sqrt(10) ) / 2 is negative, and u = ( -sqrt(2) + sqrt(10) ) / 2 is positive. u = x^2, so u can only be positive. Thus the only solutions are the solutions to the equation come from x^2 = ( -sqrt(2) + sqrt(10) ) / 2. The solutions are x = sqrt( ( -sqrt(2) + sqrt(10) ) / 2 ) and x = -sqrt( ( -sqrt(2) + sqrt(10) ) / 2 ). Approximations to three significant figures are x = .935 and x = -.935. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think it seems that the a variable is not present in your given solution and a sqrt root that im not sure where it came from. The way I understand to plug in the variables would be -sqrt(2)(+)(-)sqrt(sqrt(2)^2-4*u^2*-2/2*u^2. Perhaps you can show me what I have done wrong. ------------------------------------------------ Self-critique Rating:2 Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: