document.write( "Question 584741: how to get the solution of this equation. 2^2x+2^x-1=0 \n" ); document.write( "
Algebra.Com's Answer #372865 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
2^2x is the same as (2^x)^2
\n" ); document.write( "your equation of:
\n" ); document.write( "2^2x + 2^x - 1 = 0 becomes:
\n" ); document.write( "(2^x)^2 + (2^x) - 1 = 0
\n" ); document.write( "unfortunately, this doesn't factor.
\n" ); document.write( "it does have real roots though, so you can factor it using the quadratic formula.
\n" ); document.write( "using the quadratic formula, i get:
\n" ); document.write( "2^x = -1.6180339887
\n" ); document.write( "or:
\n" ); document.write( "2^x = .6180339887
\n" ); document.write( "to solve for x, we take the log of both sides of each equation to get:
\n" ); document.write( "log(2^x) = log(-1.618033989)
\n" ); document.write( "by the law of logarithms that states that log(x^a) = a*log(x), we get:
\n" ); document.write( "x*log(2) = log(-1.618033989)
\n" ); document.write( "unfortunately, we can't get the log of a negative number, so this solution is invalid.
\n" ); document.write( "we still have our other solution to test though.
\n" ); document.write( "that one is:
\n" ); document.write( "2^x = .6180339887
\n" ); document.write( "we take the log of both sides of this equation to get:
\n" ); document.write( "log(2^x) = log(.6180339887)
\n" ); document.write( "by the laws of logarithms again, we get:
\n" ); document.write( "x*log(2) = log(.6180339887)
\n" ); document.write( "we divide both sides of this equation by log(2) to get:
\n" ); document.write( "x = log(.6180339887) / log(2) which we can solve to get:
\n" ); document.write( "x = -.6942419136
\n" ); document.write( "we substitute that value of x in our original equation of:
\n" ); document.write( "2^(2x) + 2^x - 1 = 0 to get:
\n" ); document.write( "2^(2*(-.6942419136)) + 2^(-.6942419136) - 1 = 0
\n" ); document.write( "solve using your calculator to get:
\n" ); document.write( "0 = 0
\n" ); document.write( "this confirms the value of -.6942419136
\n" ); document.write( "the key to this problem was that 2^2x is the same as (2^x)^2.
\n" ); document.write( "our variable becomes (2^x) and we solve just like we solve any other quadratic equation except we then have to solve to find the exponent of x. \n" ); document.write( "
\n" );