document.write( "Question 1195493: Let a and B be solutions of the quadratic equation $\"x%5E2%2Bbx%2B3=0\"$ . Find all values of b such that $\"a%5E2%2BB%5E2=3\"$ .\r
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Algebra.Com's Answer #827993 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Let a and B be solutions of the quadratic equation x^2 + bx + 3 = 0.
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document.write( "First, we have this identity\r\n" );
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document.write( "    a^2 + B^2 = (a + B)^2 - 2aB.    (1)\r\n" );
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document.write( "In this identity, we replace left side  a^2 + B^2 by 3;\r\n" );
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document.write( "next, we replace (a+B) by -b, according to Vieta's theorem;\r\n" );
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document.write( "and replace aB by 3, using the Vieta's theorem again.\r\n" );
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document.write( "We get then from (1)\r\n" );
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document.write( "    3 = (-b)^2 - 2*3,\r\n" );
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document.write( "which gives\r\n" );
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document.write( "    3 + 6 = b^2,\r\n" );
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document.write( "or \r\n" );
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document.write( "    b^2 = 9.\r\n" );
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document.write( "ANSWER.  b^2 must be 9;  so \"b\" may have values 3 or -3.\r\n" );
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document.write( "As the last step of the solution, we chould check the problem's statement for b = 3 and b = -3.\r\n" );
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document.write( "It means \r\n" );
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document.write( "    - (a) to find the solutions to equation x^2 + 3x + 3 = 0 and to check that \r\n" );
document.write( "          the sum of their squares is 3;\r\n" );
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document.write( "    - (b) to find the solutions to equation x^2 - 3x + 3 = 0 and to check that \r\n" );
document.write( "          the sum of their squares is 3.\r\n" );
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document.write( "Both steps are simple arithmetic, so I leave it for you.\r\n" );
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document.write( "After completing the check, we can state for sure that \"b\" may have two possible values 3 and -3.\r\n" );
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