document.write( "Question 1210634: Fill in the blanks to make a true statement.
\n" ); document.write( "___ + ____i * 2 = ___ + 60i + ___ + ___ i \n" ); document.write( "

Algebra.Com's Answer #854578 by KMST(5382) $\"\"$ $\"About$

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This is sort of a riddle to make a student think of an answer, but there are many possible answers.
\n" ); document.write( "I was introduced to complex numbers in math as a 7th grader and much later reintroduced to them again as a college student.
\n" ); document.write( "This question could be from a young student, new to algebra, so I will try not to use many mathematical terms that such as student would not have heard before.
\n" ); document.write( "There are 3 complex numbers in that equation, represented as the \"___ + ____i\" before *2 , ___ + 60i, and the other ___ + ___ i.
\n" ); document.write( "All those blanks are to be filled with numbers that could all be different.
\n" ); document.write( "I would like to name the blanks $\"a\"$ , $\"b\"$ , $\"c\"$ , $\"d\"$ , and $\"e\"$ , in that order, to be able to tell those numbers apart.
\n" ); document.write( "If ___ + ____i is a complex number, then ___ + ____i*2 must mean the whole ___ + ____i complex number is multiplied times 2, not just its imaginary part.
\n" ); document.write( "Then, when filling the blanks with the letters above, I write
\n" ); document.write( " $\"%28a%2Bbi%29%2A2=%28c%2B60i%29%2B%28d%2Bei%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "I have to work with the real parts (a, c, and d) of those numbers separate from the imaginary parts d, 60, and e.
\n" ); document.write( "Then the problem becomes a set of equations that have to be made true at the same time, what we call a system of equations.
\n" ); document.write( " $\"%28a%2Bbi%29%2A2=%28c%2B60i%29%2B%28d%2Bei%29\"$ --> $\"2a%2B2bi=c%2B60i%2Bd%2Bei\"$ --> $\"2a%2B2bi=%28c%2Bd%29%2B%2860%2Be%29i\"$ --> $\"system%282a=c%2Bd%2C2b=60%2Be%29\"$
\n" ); document.write( "There are only 2 equations in the system $\"system%282a=c%2Bd%2C2b=60%2Be%29\"$ to determine the 5 numbers a, b, c, d, and e.
\n" ); document.write( "Of course, 2 equations is not enough to determine a set of 5 numbers, so there will be many posible answers.
\n" ); document.write( "Any of the infinte number of solutions to that system of equations will make the statement true.
\n" ); document.write( "To make it simple I could choose $\"a=b=1\"$ as part of the solution.
\n" ); document.write( "Then, substituting those values for a and b, I get $\"system%282=c%2Bd%2C2=60%2Be%29\"$ .
\n" ); document.write( "I still have many choices to finish finding a single set of values for my solution to that riddle.
\n" ); document.write( "For a simple solution, I could choose $\"c=d=1\"$ as a solution to $\"2=c%2Bd\"$ .
\n" ); document.write( "After that, I just need to solve $\"2=60%2Be\"$ --> $\"e=-58%29\"$ to have $\"system%28a=b=c=d=1%2Ce=-58%29\"$ as one solution to the problem,
\n" ); document.write( "and I could complete the blanks as
\n" ); document.write( " $\"%22___1__%22%2B%22__1__%22i+%2A2=%22__1__%22%2B60i%2B%22__1__%22%2B%22__1__%22i\"$ \n" ); document.write( "

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