document.write( "Question 481847: 2^(X)+2^(4-x)=17 \n" ); document.write( "

Algebra.Com's Answer #330922 by jsmallt9(3758) $\"\"$ $\"About$

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$\"2%5Ex%2B2%5E%284-x%29=17\"$
\n" ); document.write( "One way to solve this would be to start be rewriting the equation in terms of $\"2%5Ex\"$ . We can use the division rule for exponents, in reverse, on the second term to get:
\n" ); document.write( " $\"2%5Ex%2B2%5E4%2F2%5Ex=17\"$
\n" ); document.write( "The numerator simplifies to 16:
\n" ); document.write( " $\"2%5Ex%2B16%2F2%5Ex=17\"$
\n" ); document.write( "Now we can simplify the equation my eliminating the fraction. This can be done by multiplying both sides of the equation by $\"2%5Ex\"$ :
\n" ); document.write( " $\"2%5Ex%2A%282%5Ex%2B16%2F2%5Ex%29=2%5Ex%2A%2817%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"2%5E%282x%29+%2B+16=17%2A2%5Ex\"$
\n" ); document.write( "Next we get a zero on the right side by subtracting that term from each side:
\n" ); document.write( " $\"2%5E%282x%29+-+17%2A2%5Ex+%2B+16=0\"$
\n" ); document.write( "Since the exponent of $\"2%5E%282x%29\"$ is twice the exponent of the exponent in the middle term, $\"2%5Ex\"$ , this equation is in what is called \"quadratic form\". These kinds of equations can be solved just like \"regular\" quadratic equations. To make it easier to understand I am going to use a temporary variable:
\n" ); document.write( "Let $\"q+=+2%5Ex\"$
\n" ); document.write( "The $\"q%5E2+=+%282%5Ex%29%5E2+=+2%5E%282x%29\"$
\n" ); document.write( "Substituting these into the equation we get:
\n" ); document.write( " $\"q%5E2+-+17q+%2B16+=+0\"$
\n" ); document.write( "This obviously a quadratic equation and factoring it is easy:
\n" ); document.write( " $\"%28q-1%29%28q-16%29+=+0\"$
\n" ); document.write( "By the Zero Product Property:
\n" ); document.write( "q-1 = 0 or q-16 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "q = 1 or q = 16
\n" ); document.write( "We have solved for q. But we want to solve for x. Now we substitute for the temporary variable:
\n" ); document.write( " $\"2%5Ex+=+1\"$ or $\"2%5Ex+=+16\"$
\n" ); document.write( "Only one power of 2 is equal to 1 and that is zero. So x = 0 is the only solution for the first equation. And there is only one power of 2 that is 16 and that is 4. So the only solution to the second equation is x = 4. Putting these together the complete solution to your equation is:
\n" ); document.write( "x = 0 or x = 4

\n" ); document.write( "Once you have done a few \"quadratic form\" problems like this you will no longer need the temporary variable. You will see how to go from:
\n" ); document.write( " $\"2%5E%282x%29+-+17%2A2%5Ex+%2B+16=0\"$
\n" ); document.write( "to
\n" ); document.write( " $\"%282%5Ex+-1%29%282%5Ex+-+16%29+=+0\"$
\n" ); document.write( "to
\n" ); document.write( " $\"2%5Ex+-+1+=+0\"$ or $\"2%5Ex+-+16+-+0\"$
\n" ); document.write( "etc. \n" ); document.write( "

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