document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=498215>Question 498215</A>: <I><SPAN STYLE=\"word-break: break-all;\"> which is a possible solution of <BR>\n" );
document.write( "6 + sqrt(3x^4 + 1) = 10 - 2x^2\r<BR>\n" );
document.write( "\n" );
document.write( "the answer is  -sqrt(15) but i don't know the solution </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #336975 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=336975\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the answer is:\r<BR>\n" );
document.write( "\n" );
document.write( "possible solutions to the given equation are:<BR>\n" );
document.write( "x = +/- 1<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = +/- sqrt(15)<BR>\n" );
document.write( "here's how this equation is solved:<BR>\n" );
document.write( "original equation is:<BR>\n" );
document.write( "6 + sqrt(3x^4 + 1) = 10 - 2x^2<BR>\n" );
document.write( "subtract 6 from both sides of the equation to get:<BR>\n" );
document.write( "sqrt(3x^4 + 1) = 4 - 2x^2<BR>\n" );
document.write( "square both sides of the equation to get:<BR>\n" );
document.write( "3x^4 + 1 = (4 - 2x^2)^2<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "3x^4 + 1 = 4x^4 - 16x^2 + 16<BR>\n" );
document.write( "subtract 3x^4 from both sides of the equation and subtract 1 from both sides of the equation to get:<BR>\n" );
document.write( "0 = x^4 - 16x^2 + 15<BR>\n" );
document.write( "this is the same as:<BR>\n" );
document.write( "x^4 - 16x^2 + 15 = 0<BR>\n" );
document.write( "let y = x^2 to get:<BR>\n" );
document.write( "y^2 - 16y + 15 = 0<BR>\n" );
document.write( "this is a quadratic equation that can be factored with:<BR>\n" );
document.write( "(y - 1) * (y - 15) = 0<BR>\n" );
document.write( "this gets:<BR>\n" );
document.write( "y = 1<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "y = 15<BR>\n" );
document.write( "since y = x^2, then this is the same as:<BR>\n" );
document.write( "x^2 = 1<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x^2 = 15<BR>\n" );
document.write( "take the square root of both sides of these equations and you get:<BR>\n" );
document.write( "x = +/- 1<BR>\n" );
document.write( "x = +/- sqrt(15)<BR>\n" );
document.write( "these are possible solutions of the original equation.<BR>\n" );
document.write( "this does not mean that they are actual solutions to the original equation.<BR>\n" );
document.write( "to see if we did the factoring right, we substitute in the revised equations that got us to this point.<BR>\n" );
document.write( "that equation is:<BR>\n" );
document.write( "3x^4 + 1 = (4 - 2x^2)^2<BR>\n" );
document.write( "if we substitute + 1 for x in that equation, we get 0 as we should.<BR>\n" );
document.write( "if we substitute - 1 for x in that equation, we get 0 as we should.<BR>\n" );
document.write( "if we substitute sqrt(15) for x in that equation, we get 0 as we should.<BR>\n" );
document.write( "if we substitute sqrt(15) for x in that equation, we get 0 as we should.<BR>\n" );
document.write( "this means that our solution is good for the equation that we evaluated it at.<BR>\n" );
document.write( "we need, however, to confirm that these  are solutions to the original equation.<BR>\n" );
document.write( "the original equation is:<BR>\n" );
document.write( "6 + sqrt(3x^4 + 1) = 10 - 2x^2<BR>\n" );
document.write( "if this equation is equal, then we can subtract (10-2x^2) from both sides of the equation to get:<BR>\n" );
document.write( "6 + sqrt(3x^4 + 1) - (10 - 2x^2) = 0<BR>\n" );
document.write( "if we substitute + 1 for x in that equation, we get 0 as we should.<BR>\n" );
document.write( "if we substitute - 1 for x in that equation, we get 0 as we should.<BR>\n" );
document.write( "if we substitute + sqrt(15) for x in that equation, we get 52 which we should not have gotten if sqrt(15) was a solution.  apparently it is not.<BR>\n" );
document.write( "if we substitute - sqrt(15) for x in that equation, we get 52 which we should not have gotten if sqrt(15) was a solution.  apparently it is not.<BR>\n" );
document.write( "while +/- sqrt(15) were possible solutions to the original equations, it turned out that they were not solutions to the original equations.<BR>\n" );
document.write( "while +/- 1 were possible solutions to the original equations, it turned out that they were actually solutions to the original equations.<BR>\n" );
document.write( "the answer is that the solutions to the original equation given is:<BR>\n" );
document.write( "x = +/- 1.<BR>\n" );
document.write( "if we graph the original equation and the revised equation, this is what we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-5%2C5%2C-100%2C100%2C6+%2B+sqrt%283x%5E4%2B1%29-%2810-2x%5E2%29%2C3x%5E4%2B1+-+%284-2x%5E2%29%5E2%2C900%28x-sqrt%2815%29%29%2C900%28x%2Bsqrt%2815%29%29%2C900%28x-1%29%2C900%28x%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can see that the revised equation has roots at +/- 1 and +/- sqrt(15).<BR>\n" );
document.write( "you can also see that the original equation only has roots at +/- 1.<BR>\n" );
document.write( "that is why +/- 1 was a solution to the original equation, but +/- sqrt(15) was not.<BR>\n" );
document.write( "in this graph, there are vertical lines at x = +/- sqrt(15) and x = +/- 1 to show you that the revised equation has roots in both places (y = 0) while the original equation only has roots at x = +/- 1 and does not have roots at y = +/- sqrt(15).<BR>\n" );
document.write( "the fact that the graph of the revised equation and the graph of the original equation intersect at some other point besides the 0 point is meaningless to the problem.<BR>\n" );
document.write( "the problem was to find the solution to the original equation.<BR>\n" );
document.write( "if we had graphed the original equation by itself, we would only have seen roots at +/- 1.<BR>\n" );
document.write( "that was sufficient to solve the problem.<BR>\n" );
document.write( "the fact that we created the revised equation to solve the original equation was a matter of convenience to make a 4th degree equation equivalent to a second degree equation, which we know how to solve because it's a quadratic equation.  <BR>\n" );
document.write( "we got possible solutions, but there was no guarantee that a solution to the revised equation would also be a solution to the original equation.  we did preserve the equality, but to say that these 2 equations are identical would not be correct.<BR>\n" );
document.write( "you get a possible solution, but you still have to confirm that the possible solution is an actual solution by substituting that value for x in the original equation.<BR>\n" );
document.write( "to summarize:<BR>\n" );
document.write( "+/- 1 and +/- sqrt(15) were possible solutions to the original equation.<BR>\n" );
document.write( "+/- 1 were the only solutions to the original equation.<BR>\n" );
document.write( "that, to the best of my knowledge, is your answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );