document.write( "Question 1163110: For how many integers m is it true that x^2 −8+(4m)(x−2)^2 +(m^2)((x−2)^2) ≥0 for all real values of x?\r
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Algebra.Com's Answer #787105 by Edwin McCravy(20054) $\"\"$ $\"About$

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document.write( "For a parabola with equation y=ax²+bx+c to have non-negative values for all\r\n" );
document.write( "x, then except possibly for its vertex, it must be totally above the x-axis,\r\n" );
document.write( "or possibly with only its vertex on the x-axis with the parabola tangent to\r\n" );
document.write( "the x-axis at its vertex.  That is,\r\n" );
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document.write( "1. The parabola must open upward, i.e., its leading coefficient is positive.\r\n" );
document.write( "2. It must have only one or no real zeros, i.e., its discriminant is \r\n" );
document.write( "non-positive.  That is, a > 0 and b²-4ac ≤ 0.\r\n" );
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document.write( "Requiring that the leading coefficient must be positive:\r\n" );
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document.write( "That represents a parabola opening upward. It has approximately zeros -3.7\r\n" );
document.write( "and -0.3, thus for integer m,\r\n" );
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document.write( "Requiring that the discriminant be non-positive:\r\n" );
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document.write( "Divide through by 16\r\n" );
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document.write( "That is a parabola opening upward, and has approximate zeros -3.4 and -0.6,\r\n" );
document.write( "thus for integer m,\r\n" );
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document.write( "This requirement for m contradicts the requirement that\r\n" );
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document.write( "Thus there are no solutions.\r\n" );
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document.write( "Edwin

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