document.write( "Question 1108169: Find the square roots of z=5−12i \n" ); document.write( "

Algebra.Com's Answer #723248 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "Suppose that a+bi is a square root of 5 - 12i. \r\n" );
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document.write( "Then, (a+bi)^2 = (a^2 - b^2) + (2ab)i = 5 - 12i. \r\n" );
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document.write( "Equate real and imaginary parts: \r\n" );
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document.write( "a^2 - b^2 = 5 \r\n" );
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document.write( "2ab = -12 ==> b = -6/a. \r\n" );
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document.write( "So, a^2 - (-6/a)^2 = 5 \r\n" );
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document.write( "==> a^2 - 36/a^2 = 5 \r\n" );
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document.write( "==> a^4 -5a^2 - 36 = 0. \r\n" );
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document.write( "==> (a^2 -9)(a^2 + 4) = 0. \r\n" );
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document.write( "Since a must be real, a = 3 or -3. \r\n" );
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document.write( "This gives b = 2 or -2, respectively. \r\n" );
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document.write( "Thus, we have two square roots: 3-2i or -3+2i.\r\n" );
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\n" ); document.write( "Be aware: the final answer by the other tutor is incorrect !\r
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\n" ); document.write( "On complex numbers, there are the lessons in this site\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number (*)\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - A curious example of an equation in complex numbers which HAS NO a solution\r
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\n" ); document.write( "\n" ); document.write( "            ------>>>   Notice that your problem is Problem 1 of the lesson marked (*) in the list.   <<<------\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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