document.write( "Question 80076: What values of b in the following equations will give one or more real number solutions?\r
\n" ); document.write( "\n" ); document.write( "a). 3x^2+bx-3=0
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\n" ); document.write( "\n" ); document.write( "what is a rule for judging if an equation has solutions by looking at it in standard form?\r
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Algebra.Com's Answer #57434 by Edwin McCravy(20055) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "What values of b in the following equations will\r\n" );
document.write( " give one or more real number solutions?\r\n" );
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document.write( "a). 3x^2+bx-3=0\r\n" );
document.write( "b). 5x^2+bx+1=0\r\n" );
document.write( "c). -3x^2+bx-3=0\r\n" );
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document.write( "what is a rule for judging if an equation has \r\n" );
document.write( "solutions by looking at it in standard form?\r\n" );
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document.write( "thanks!\r\n" );
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document.write( "When we have an equation of the form \r\n" );
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document.write( "ax² + bx + c = 0\r\n" );
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document.write( "we form the dicriminant \r\n" );
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document.write( "Discriminant = b² - 4ac\r\n" );
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document.write( "The rule is\r\n" );
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document.write( "There will be at least one real solution \r\n" );
document.write( "if and only if the discriminant is greater \r\n" );
document.write( "than or equal 0. There will be one real\r\n" );
document.write( "solution if the discriminant equals 0 and \r\n" );
document.write( "2 real solutions if the discriminant is \r\n" );
document.write( "greater than 0.\r\n" );
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document.write( "a).     3x² + bx - 3 = 0\r\n" );
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document.write( "       a = 3, c = -3\r\n" );
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document.write( "       b² - 4ac > 0\r\n" );
document.write( "  b² - 4(3)(-3) > 0\r\n" );
document.write( "        b² + 36 > 0\r\n" );
document.write( "             b² > -36\r\n" );
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document.write( "The square of every real number is\r\n" );
document.write( "always positive or zero and therefore \r\n" );
document.write( "is always greater than -36\r\n" );
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document.write( "Answer: b can be any real number.  \r\n" );
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document.write( "b). 5x² + bx + 1 = 0\r\n" );
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document.write( "       a = 5, c = 1\r\n" );
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document.write( "       b² - 4ac > 0\r\n" );
document.write( "   b² - 4(5)(1) > 0\r\n" );
document.write( "        b² - 20 > 0\r\n" );
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document.write( "(I will use V for the square root radical)\r\n" );
document.write( "                      __      _\r\n" );
document.write( "critical values are ±V20 = ±2V5 roughly 4.4 and -4.47\r\n" );
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document.write( "------------o----------------o--------------\r\n" );
document.write( "          -2V5              2V5\r\n" );
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document.write( "Substituting a test value below -2V5, b=-5\r\n" );
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document.write( "     (-5)² - 20 > 0\r\n" );
document.write( "        25 - 20 > 0\r\n" );
document.write( "              5 > 0\r\n" );
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document.write( "That's true, so we shade the region\r\n" );
document.write( "of the number line left of -2V5.\r\n" );
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document.write( "<===========o----------------o--------------\r\n" );
document.write( "          -2V5              2V5  \r\n" );
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document.write( "Substituting a test value between -2V5 and 2V5, b=0\r\n" );
document.write( "      \r\n" );
document.write( "      0² - 20 > 0\r\n" );
document.write( "       0 - 20 > 0\r\n" );
document.write( "          -20 > 0\r\n" );
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document.write( "That's false, so we do not shade the region\r\n" );
document.write( "of the number line between them.\r\n" );
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document.write( "Substituting a test value above 2V5, b=5\r\n" );
document.write( "      \r\n" );
document.write( "      (5)² - 20 > 0\r\n" );
document.write( "        25 - 20 > 0\r\n" );
document.write( "              5 > 0\r\n" );
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document.write( "That's true, so we shade the region\r\n" );
document.write( "of the number line right of 2V5.\r\n" );
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document.write( "<===========o----------------o=============>\r\n" );
document.write( "          -2V5              2V5  \r\n" );
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document.write( "So the values of b for which the solutions are\r\n" );
document.write( "real are\r\n" );
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document.write( "         (-oo,-2V3) U (2V3,oo)\r\n" );
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document.write( "c). -3x^2+bx-3=0\r\n" );
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document.write( "       a = -3, c = -3\r\n" );
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document.write( "       b² - 4ac > 0\r\n" );
document.write( " b² - 4(-3)(-3) > 0\r\n" );
document.write( "        b² - 36 > 0\r\n" );
document.write( " (b - 6)(b + 6) > 0 \r\n" );
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document.write( "                           \r\n" );
document.write( "critical values are 6 and -6\r\n" );
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document.write( "------------o----------------o--------------\r\n" );
document.write( "           -6                6 \r\n" );
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document.write( "Substituting a test value below -6, b = -7\r\n" );
document.write( "      \r\n" );
document.write( "     (-7)² - 36 > 0\r\n" );
document.write( "        49 - 36 > 0\r\n" );
document.write( "             13 > 0\r\n" );
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document.write( "That's true, so we shade the region\r\n" );
document.write( "of the number line left of -6.\r\n" );
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document.write( "<===========o----------------o--------------\r\n" );
document.write( "           -6                6  \r\n" );
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document.write( "Substituting a test value between -6 and 6, b=0\r\n" );
document.write( "      \r\n" );
document.write( "      0² - 36 > 0\r\n" );
document.write( "       0 - 36 > 0\r\n" );
document.write( "          -36 > 0\r\n" );
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document.write( "That's false, so we do not shade the region\r\n" );
document.write( "of the number line between them.\r\n" );
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document.write( "Substituting a test value above 6, b=7\r\n" );
document.write( "      \r\n" );
document.write( "      (7)² - 36 > 0\r\n" );
document.write( "        49 - 36 > 0\r\n" );
document.write( "             13 > 0\r\n" );
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document.write( "That's true, so we shade the region\r\n" );
document.write( "of the number line right of 6.\r\n" );
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document.write( "<===========o----------------o=============>\r\n" );
document.write( "           -6                6  \r\n" );
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document.write( "So the values of b for which the solutions are\r\n" );
document.write( "real are\r\n" );
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document.write( "         (-oo,-6) U (2V3,6)\r\n" );
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document.write( "Edwin
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