How to solve x^2 = 2^x analytically? (and get all the answers)?
464950
I believe that you will need to use Numeriacl Analysis to solve an equation such as this. It has been many, many years since I was a student in Math but I recall that your approach seems to be the way to go. Your method of graphing was a good analytic approach. Next using the successive approx was good. \n\nAs an alternative, You could try to expand 2^x- x^2=0 into a series and see if there is an answer in closed form. You will have to show limits etc.  I think that you will discover that you can't solve this in closed form. The equation that you have to solve is:\nx^2-exp(x*ln(2))=0\n\nAn example of functions that can't get a closed form answer is trying to integrate this function: \nf(x)=exp(x^2). You will discover that you can Numerically integrate the exp(x^2) but can't get an answer in Closed Form.\n\nMy conclusion is that you can't solve the third solution in closed form (analytically). The third answer seems to be bwtween -1 and -1/2. I did this quickly in my head. Your graph probably shows this if my calculations in my head are correct. So, if you try to solve this using one of the Numerical Analysis Methods, start the first approx with maybe -.75. You can do this with a Calculator. If you have Fortran or basic on your PC, you can write a simple program. When I went to school, we used Fortran & Basic. Is that used today? Or do you use Pascal? Well, you can write a simple program to solve this using your Computer.
904672
let u = x^2 + 1, then you have du = 2xdx, => dx = 1/2x du,\n\nsubstitute back you are on the way  to solve it
747583
y= √(1/(x² + 1) – 2) \n\nWell 1/(x² + 1) – 2 ≥ 0\nie 1/(x² + 1) ≥ 2\nSince x² + 1 ≥ 0 for all x\nOn taking reciprocals\nx² + 1 ≤ ½\nx² ≤ -½ This is not possible as x² ≥ 0 for all x \nSo there is no domain\n----------------------- -----------------------\nWell that makes sense as  x² + 1 ≥ 1\nSo 1/(x² + 1) ≤ 1\nSo 1/(x² + 1) - 2 ≤ -1\n\nSo taking square roots gives no real solutions.
791534
(2asinx*cosx)^2 + a^2(cos^2x - sin^2x)^2\n=4a^2 sin^2xCos^2x + a^2(cos^4x+sin^4x -2sin^2xCos^2x )\n=2a^2 sin^2xCos^2x + a^2cos^4x+a^2sin^4x\n=a^2(cos^2x + sin^2x)^2\n=a^2
844053
I'll rewrite the question for you, because we usually denote ^ as "to the power of" for simplicity.\n\nx^2 + y^2 - 4x + 2y = 4\n\nWe have to do a process called completing the square, for both the x and y variables.  So first, let's group all the x terms together and y terms together.\n\nx^2 - 4x + y^2 + 2y = 4\n\nTo complete the square, we have to add 4 to the x terms and 1 to the y terms.  Since we're adding 4 and 1 to the left hand side, we have to do the same to the right hand side.\n\nx^2 - 4x + 4 + y^2 + 2y + 1 = 4 + 4 + 1\n\nNow, we factor those now square polynomials.\n(x-2)^2 + (y+1)^2 = 9\n\nThe center of the circle is given by the values in the brackets, but the negative of them.  In the above case, the center is at (2,-1), since you take the negative of -2 (the value next to the x) and 1 (the value next to the y).\n\nTo get the radius, you take the square root of the right hand side, so the square root of 9 is 3.
699436
(2^x-2^-x) / 3 (the whole thing is divided by 3)\n\nThat is not an equation to be solved.   So I don't know where you are getting an equals sign (=).??\n\nYou may be able to simplify it, but you cannot say x= something unless you were provided with an EQUATION.\n\nI will do some work on it.\n\nAfter further review, try to input your value for x into the statement\n(2^x-2^-x) / 3 and what you get is another question.   Your value then EQUALS what?   Does this answer your question?    All you will get is a value.
551906
Add each equation\nA + B = 3x^2 + 2x - 2\nA - B = -x^2 + 4x - 8\n2A = 2x^2 + 6x - 10\nDivide each side by 2\nA = x^2 + 3x - 5\nNow plug into one of the original equations to get B\nB = 3x^2 + 2x - 2 - (x^2 + 3x - 5)\nB = 2x^2 - x + 3
931054
Group all elements with y together:\n\ny(x^2 + 6x +9) -x^2 + 2x + 3 = 0\n\nMove the rest to the other side:\n\ny(x+3)^2 = x^2 - 2x + 3 (1)\n\nx = -3 is obviously not the solution of this equation therefore we can divide both sides of (1) by (x+3)^2.\n\n(1) is equivalent to:\n\ny = (x^2 - 2x + 3) / (x+3)^2
959661
if by this you mean\n\nx^2 - 2x - 2 = 0\n\nx^2 - 2x = 2\nx^2 - 2x + 1 = 3\n(x - 1)^2 = 3\nx - 1 = sqrt(3)\nx = 1 ± sqrt(3)\n\n-------------------------\n\nbut if you mean\n\n7x^2 - 2x - 2 = 0\n7x^2 - 2x = 2\nx^2 - (2/7)x = (2/7)\nx^2 - (2/7)x + (1/49) = (63/49)\n(x - (1/7))^2 = (9/7)\nx - (1/7) = (3/7)sqrt(7)\nx = (1/7) ± (3/7)sqrt(7)\nx = (1/7)(1 ± 3sqrt(7))
765513
do u mean 6x^2(-9x^3) = -54x^5\nmultiply the constants and add the exponents
394036
Try to solve the eqs with quad rule:\na*x^2 + b*x + c=0\nx=(-b +- Sqrt(b^2-4*a*c))/(2*a)\n\nx^2+2x+5=0 => Get two Complex answers, \nso no it can't be factorized.\n\nx^2+4x+5=0 => Get two Complex answers, \nso no it can't be factorized.\n\nx^2+4x+3=(x+3)*(x+1)=0; \nSo, yes, this can be factorized. \nSolutions are x=-3 and x= -1\n\nx^2+2x-3=(x+3)*(x-1)=0;\nSo, yes, this can be factorized. \nSols are x= -3 and x=1
764068
2x - 2 = x^2 - 4x + 3\nSet one side equal to 0\n0 = x^2 - 6x + 5\nFactor\n0 = (x - 5)(x - 1)\nSet each factor equal to 0\n0 = x - 5              0 = x - 1\nSolve\n5 = x                  1 = x
57070
Starting with\nx^2 - 5x - 6.25 = 0\n\nMOVE the lone term to the right side of the equation.\n\nSo\n x^2 - 5x = 6.25\n\nTo comlete the square on the left:\nDIVIDE the coefficient of the x term (5) by two (2.5).  Then SQUARE it.\n\nADD \n 2.5^2 (which happens to be 6.25) to both sides\n\nNow\n x^2 - 5x + 6.25 = 6.25 + 6.25\n\nREWRITE the left which is now a perfect square\n (x-2.5)^2 = 12.5 (or 25/2)\n\nTAKE THE SQUARE ROOT OF BOTH SIDES\n x - 2.5 = +or- Square root of 12.5\n\nANSWER\n x = Sqrt 12.5 + 2.5   -or-\n x = 2.5 - Sqrt 12.5
681236
Multiply each term in the first () by each term in the second ():\n\n2x times 2x = 4x^2\n2x times 1 = 2x\n-3 times 2x = -6x\n-3 times 1 = -3\n\nThen combine 2x and -6x\nAnswer: 4x^2 - 4x - 3
263286
the problem you presented above can be represented as,\n\n(8x^8/2x^2) + (16x^6/2x^2) - (12x^3/2x^2)\n\nthen you can simply divide each of the term\nyou'll get\n\n4x^6 + 8x^4 - 6x
640891
I'm guessing you meant 17x^3 - 10x^2 - 6x +4 and 2x^2- 6x +2\nIf that's right, you just add like powers of x and their coefficients to get 17x^3 - 8x^2 -12x +6.
983225
remember the rule nx^n-1\n2x^2-1\n \nSo in this case it is 2x