document.write( "Question 1208090: If (1 + a)(1 + b)(a + b) = 2022
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Algebra.Com's Answer #846259 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "If (1 + a)(1 + b)(a + b) = 2022 and a^3 + b^3 = 1993, find (a + b).
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document.write( "First,  (1 + a)(1 + b)(a + b) = (1 + a + b + ab)*(a+b) = (a+b) + (a+b)^2 + ab*(a+b).\r\n" );
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document.write( "Next,  a^3 + b^3 = (a+b)*(a^2 - ab + b^2) = (a+b)*((a+b)^2-3ab) = (a+b)^3 - 3ab*(a+b).\r\n" );
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document.write( "Thus you have these two equations\r\n" );
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document.write( "    (a+b) + (a+b)^2 + ab*(a+b)  = 2022    (1)\r\n" );
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document.write( "    (a+b)^3         - 3ab*(a+b) = 1993    (2)\r\n" );
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document.write( "Multiply equation (1) by 3; keep equation (2) as is.  Then add them.  You will get\r\n" );
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document.write( "    3(a+b) + 3(a+b)^2 + (a+b)^3 = 3*2022 + 1993,\r\n" );
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document.write( "or\r\n" );
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document.write( "    3(a+b) + 3(a+b)^2 + (a+b)^3 = 8059.\r\n" );
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document.write( "Add 1 to both sides\r\n" );
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document.write( "    1 + 3(a+b) + 3(a+b)^2 + (a+b)^3 = 8060.\r\n" );
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document.write( "Notice that the left side is ((a+b)+1)^3.\r\n" );
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document.write( "So, the last equation is\r\n" );
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document.write( "    ((a+b)+1)^3 = 8060.\r\n" );
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document.write( "Take cube root of both sides\r\n" );
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document.write( "    (a+b)+1 = .\r\n" );
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document.write( "From this, get\r\n" );
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document.write( "     a+b =  -  = 19.04987552  (rounded).\r\n" );
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document.write( "ANSWER.  If (1 + a)(1 + b)(a + b) = 2022 and a^3 + b^3 = 1993,  then  a+b =  -  = 19.04987552  (rounded).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus you have both the exact formula and an approximate value.\r
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