document.write( "Question 1078806: Find all the values of n such that x^2 + bx + (b/2)^2 = n has two real roots. Thank you tutors. \n" ); document.write( "
Algebra.Com's Answer #693241 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "A quadratic equation with 0 on the right has two\r\n" );
document.write( "real roots if the discriminant > 0\r\n" );
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document.write( "We find the discriminant which is \r\n" );
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document.write( "here a=1, b=b, c=\r\n" );
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document.write( "So the discriminant is\r\n" );
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document.write( " which we set \r\n" );
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document.write( "Multiply through by 4\r\n" );
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document.write( "Factor out b²\r\n" );
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document.write( "The critical numbers are 0,2 and -2.\r\n" );
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document.write( "[The critical numbers all cause the discriminant to = 0,\r\n" );
document.write( "which gives the quadratic only one real solution.\r\n" );
document.write( "So none of the critical numbers are possible values \r\n" );
document.write( "for b.\r\n" );
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document.write( "We place the critical numbers on a number line\r\n" );
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document.write( "---------o-------o-------o--------\r\n" );
document.write( "-4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "We test values in each interval:\r\n" );
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document.write( "Using test value -3 for the first (leftmost) interval\r\n" );
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document.write( "That's true so we shade the interval (-2,0).\r\n" );
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document.write( "---------o=======o-------o--------\r\n" );
document.write( "-4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "Using test value 1 for the next interval\r\n" );
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document.write( "That's true so we also shade the interval (0,2).\r\n" );
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document.write( "---------o=======o=======o--------\r\n" );
document.write( "-4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "Using test value 3 for the next interval\r\n" );
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document.write( "That's false.  Therefore b must be in these intervals\r\n" );
document.write( "in order for the equation to have 2 real roots:\r\n" );
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document.write( "(-2,0) U (0,2)\r\n" );
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document.write( "Edwin
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