a1, a2, a3 ,…, an  in  [-1;1] ; and (a1)^3 +(a2)^3 +...+(an)^3=0. Prove that : a1+a2+...+an <= n/3?
611271
a_i = sin(phi_i)\nsin^3(phi_i) = (3sin(phi_i) - sin(3phi_i)) / 4\nso: 0 = Sum[1..n] sin^3(phi_i) = \nSum[1..n] (3sin(phi_i) - sin(3phi_i)) / 4\nso: Sum[1..n] sin(phi_i) = 1/3 Sum[1..n] sin(3phi_i) <= n/3\nqed
913473
Are you missing a plus sign and parentheses around a+b?  If so, then\n(a^2 -2ab + b^2) = (a-b)(a-b) = (a-b)^2\nYou cannot cancel anything, so \n(a^2 -2ab + b^2)/(a+b) does not simplify.\nYou may wish to write it as (a-b)^2 /(a+b)\n\nIf you are not missing a plus sign or parentheses, then\na^2 - 2ab b^2 = a^2 - 2ab^3\nYou can factor out an a.\n= a(a-b^3)  This does not factor, so you cannot simplify.\nDivide this by a and you get a - b^3\nNow add b\nAnswer: a - 2b^3 + b
602540
x + 3 >  8 ==> x >  8 - 3  ==> x > 5\nx - 4 < -2 ==> x < -2 + 4 ==> x < 2\n\nAnswer... not possible.
286213
a) 1/(3x+3)=-7\n1=-7(3x+3)\n1=-21x-21\n1+21=-21x\n22=-21x\nx= -(22/21)\nx= -1.0476\n\nb)x-3/-1=x-2\nx-3=-1(x-2)\nx-3=-x+2\nx+x=3+2\n2x=5\nx=2.5
289224
(x+ [1+i*sqrt(3)]/2) * (x+[1-i*sqrt(3)]/2) = 0
470404
((a^2 - b^2)/8a) X ((4a^3)/(a + b)^2))\n(4a^3(a^2 - b^2))/(8a(a + b)^2)\n((4a^3)/(8a)) * ((a^2 - b^2)/(a + b)^2)\n((a^(3 - 1))/2) * (((a - b)(a + b))/((a + b)(a + b)))\n((a^2)/2) * ((a - b)/(a + b))\n\nANS : ((a^2)(a - b))/(2(a + b))\n\nOn your problem, here is where you went wrong\n\nTo make it easier, you should had kept your problem in\n\n(4a^3(a - b))/(8a(a + b)) form, then had done this\n\n((4a^3)/(8a)) * ((a - b)/(a + b))\n\n(4/8)(a^(3 - 1))((a - b)/(a + b))\n(1/2)(a^2)((a - b)/(a + b))\n((a^2)(a - b))/(a + b)
825318
(a^2 -2ab + b^2) = (a-b)(a-b) = (a-b)^2\nYou cannot cancel anything, so \n(a^2 -2ab + b^2)/(a+b) does not simplify.\n\nYou may wish to write it as (a-b)^2 /(a+b)
806157
it is \n(x/3x+1/3x)/(3/3x+x/3x)=\n=(x+1)/(x+3)
139883
(a+b)(a-b) \n= a^2 - ab + ba - b^2\n= a^2 - b^2 -ab +ab\n= a^2 - b^2
1066777
x-3=3\nx=3+3\nx=6\n\n\n15-(x)=7\n15-7=x\n8=x
895527
2x^2-8x+3x-12\n2x(x-4)+3(x-4)\n2x+3 = 0 or x-4= 0\nx = -3/2 or x = 4\n\n2x^2+10x-72\n2x^2+18x-8x-72\n2x(x+9)-8(x+9)\nx = 4 or x = -9\n\nx^2+6x-x-6\nx-1 = 0 or x+6 = 0\nx = 1 or x = -6
258025
A(n)= [1/2 (-3)^n] -1 OR A(n)= 1/2 (-3)^(n-1)\n\nAssume the first\nA(n)= [1/2 (-3)^n] -1\nA(0)= - 0.5\nA(1)= - 2.5\nA(2)= + 3.5\nA(3)= - 14.5\nA(4)= + 39.5\nA(5)= - 122.5\nA(6)= + 363.5\nA(7)= - 1094.5\nA(8)= + 3279.5  >>> this is your answer\nA(9)= - 9842.5\nA(10)= + 29523.5\n\nAssume the second\nA(n)= 1/2 (-3)^(n-1)\nA(0)= - 0.16666\nA(1)= + 0.5\nA(2)= - 1.5\nA(3)= + 4.5\nA(4)= - 13.5\nA(5)= + 40.5\nA(6)= - 121.5\nA(7)= + 364.5\nA(8)= - 1093.5  >>> this is your answer\nA(9)= + 3280.5\nA(10)= - 9841.5
705864
Alright, don't listen to the above post...\n\nGrade 12 here, answering questions all day for mere fun.\n\n12)\ny - 3 = -1/3 (x + 4)\n\nEXPLAINED\n\nadd 3 to both sides\ny = -1/3 (x + 4) + 3\n\nexpand 1/3(x + 4) \n= x * 1/3 + 4 * -1/3\n= x/3 + 4/3\ny = -1/3x - 4/3 + 3\n\nTo add -4/3 and 3, use 3 = 9/3\ny = -1/3x - 4/3 + 9/3\ny = -1/3x + 5/3\n\nTrue, my bad. I originally added 4/3 instread of -4/3\n\n\n16)\n(-2, -1) (4, -4)\n\nm = Dy / Dx\nm = ((-1) - (-4)) / ((-2) - (4))\nm = 3 / -6\nm = -1/2\n\ny = mx + b\n(-1) = -1/2(-2) + b\n-1 = 1 + b\n-1 - 1 = b\n-2 = b\n\ny = -1/2x - 2\n\nCHECK\ny = -1/2(4) - 2\ny = -2 - 2\ny = -4\nTherefore it works.\n\n18)\n(0, 0) (2, 3)\n\nm = Dy / Dx\nm = ((3) - (0)) / ((2) - (0))\nm = 3/2\n\nb = 0\n\ny = mx + b\ny = (3/2)x + 0\ny = (3/2)x\n\nNote, the lower post copied my answers
889713
3|(2x-1)/5| + 3 = 1?\nThats impossible. |anything| >= 0, so the LHS is >= 3. Did you mean something else?
390719
so you have (1/2)a+a-(6/3)=4+(1/3)a\n\nadding like terms: (3/2)a-2=4+(1/3)a\n\nsubtract (1/3)a from (3/2)a and add two to both sides:\n(3/2-1/3)a=4+2\n(9/6-2/6)a=6\n(7/6)a=6\n\nnow solve and what do you get? :)
1051369
Given a+b+c=0\na+b=-c\nCubing both sides, we get\n(a+b)^3=(-c)^3\na^3+3a^2 b+3ab^2+b^3=-c^3\na^3+b^3+c^3=-3a^2 b-3ab^2=-3ab(a+b)=-3ab(-c)=3abc\nTherefore a^3+b^3+c^3=3abc
268383
2x^2+6x-1x-3\n2x(1x+3)-1(1x+3)=0\n(2x-1)(x+3)=0\nx=1/2 or -3