document.write( "Question 1209743: If 2^x + 2^y = 6
\n" ); document.write( "and x + y = 2,
\n" ); document.write( "find 4^x + 4^y. \n" ); document.write( "
Algebra.Com's Answer #850075 by CPhill(1959)\"\" \"About 
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Here's how to solve this problem:\r
\n" ); document.write( "\n" ); document.write( "**1. Express y in terms of x:**\r
\n" ); document.write( "\n" ); document.write( "Since x + y = 2, we can write y = 2 - x.\r
\n" ); document.write( "\n" ); document.write( "**2. Substitute y in the first equation:**\r
\n" ); document.write( "\n" ); document.write( "2^x + 2^(2-x) = 6\r
\n" ); document.write( "\n" ); document.write( "**3. Simplify the equation:**\r
\n" ); document.write( "\n" ); document.write( "2^x + (2^2 / 2^x) = 6
\n" ); document.write( "2^x + (4 / 2^x) = 6\r
\n" ); document.write( "\n" ); document.write( "**4. Introduce a substitution:**\r
\n" ); document.write( "\n" ); document.write( "Let u = 2^x. Then the equation becomes:\r
\n" ); document.write( "\n" ); document.write( "u + (4/u) = 6\r
\n" ); document.write( "\n" ); document.write( "**5. Solve for u:**\r
\n" ); document.write( "\n" ); document.write( "Multiply the entire equation by u:\r
\n" ); document.write( "\n" ); document.write( "u^2 + 4 = 6u
\n" ); document.write( "u^2 - 6u + 4 = 0\r
\n" ); document.write( "\n" ); document.write( "**6. Use the quadratic formula to solve for u:**\r
\n" ); document.write( "\n" ); document.write( "u = (-b ± sqrt(b^2 - 4ac)) / 2a
\n" ); document.write( "u = (6 ± sqrt((-6)^2 - 4 * 1 * 4)) / 2 * 1
\n" ); document.write( "u = (6 ± sqrt(36 - 16)) / 2
\n" ); document.write( "u = (6 ± sqrt(20)) / 2
\n" ); document.write( "u = (6 ± 2√5) / 2
\n" ); document.write( "u = 3 ± √5\r
\n" ); document.write( "\n" ); document.write( "**7. Find the two possible values for 2^x:**\r
\n" ); document.write( "\n" ); document.write( "So, 2^x = 3 + √5 or 2^x = 3 - √5\r
\n" ); document.write( "\n" ); document.write( "**8. Find the corresponding values for 2^y:**\r
\n" ); document.write( "\n" ); document.write( "Since 2^x * 2^y = 2^(x+y) = 2^2 = 4:\r
\n" ); document.write( "\n" ); document.write( "If 2^x = 3 + √5, then 2^y = 4 / (3 + √5) = 4(3 - √5) / ((3 + √5)(3 - √5)) = 4(3 - √5) / (9 - 5) = 3 - √5
\n" ); document.write( "If 2^x = 3 - √5, then 2^y = 4 / (3 - √5) = 4(3 + √5) / ((3 - √5)(3 + √5)) = 4(3 + √5) / (9 - 5) = 3 + √5\r
\n" ); document.write( "\n" ); document.write( "Notice that the values of 2^x and 2^y are just swapped.\r
\n" ); document.write( "\n" ); document.write( "**9. Calculate 4^x + 4^y:**\r
\n" ); document.write( "\n" ); document.write( "4^x + 4^y = (2^x)^2 + (2^y)^2\r
\n" ); document.write( "\n" ); document.write( "Case 1: 2^x = 3 + √5 and 2^y = 3 - √5
\n" ); document.write( "4^x + 4^y = (3 + √5)^2 + (3 - √5)^2 = (9 + 6√5 + 5) + (9 - 6√5 + 5) = 14 + 14 = 28\r
\n" ); document.write( "\n" ); document.write( "Case 2: 2^x = 3 - √5 and 2^y = 3 + √5
\n" ); document.write( "4^x + 4^y = (3 - √5)^2 + (3 + √5)^2 = (9 - 6√5 + 5) + (9 + 6√5 + 5) = 14 + 14 = 28\r
\n" ); document.write( "\n" ); document.write( "In either case, 4^x + 4^y = 28.\r
\n" ); document.write( "\n" ); document.write( "**Final Answer:** 4^x + 4^y = 28
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