document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170004>Question 1170004</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the equation of second degree in x :<BR>\n" );
document.write( "mx²-2(m-1)x+m+1=0 where m is a non zero real parameter <BR>\n" );
document.write( "1)Following the values of m, find the existence of the roots of this equation.<BR>\n" );
document.write( "2)Find among the roots, when they exist, a relation independent of m.<BR>\n" );
document.write( "3)Using this relation, find the possible double root (x1=x2=X)<BR>\n" );
document.write( "Calculate the corresponding value of m.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851271 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851271\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the given quadratic equation step-by-step:\r<BR>\n" );
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document.write( "**1. Existence of Roots:**\r<BR>\n" );
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document.write( "The given equation is:\r<BR>\n" );
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document.write( "mx² - 2(m - 1)x + m + 1 = 0\r<BR>\n" );
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document.write( "For the roots to exist, the discriminant (Δ) must be greater than or equal to zero.\r<BR>\n" );
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document.write( "Δ = b² - 4ac\r<BR>\n" );
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document.write( "where:<BR>\n" );
document.write( "* a = m<BR>\n" );
document.write( "* b = -2(m - 1)<BR>\n" );
document.write( "* c = m + 1\r<BR>\n" );
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document.write( "Δ = [-2(m - 1)]² - 4(m)(m + 1)<BR>\n" );
document.write( "Δ = 4(m² - 2m + 1) - 4m(m + 1)<BR>\n" );
document.write( "Δ = 4m² - 8m + 4 - 4m² - 4m<BR>\n" );
document.write( "Δ = -12m + 4\r<BR>\n" );
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document.write( "For roots to exist, Δ ≥ 0:<BR>\n" );
document.write( "-12m + 4 ≥ 0<BR>\n" );
document.write( "4 ≥ 12m<BR>\n" );
document.write( "m ≤ 4/12<BR>\n" );
document.write( "m ≤ 1/3\r<BR>\n" );
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document.write( "Therefore, the roots exist when m ≤ 1/3.\r<BR>\n" );
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document.write( "**2. Relation Independent of m:**\r<BR>\n" );
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document.write( "Let x1 and x2 be the roots of the equation.\r<BR>\n" );
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document.write( "Sum of roots: x1 + x2 = -b/a = 2(m - 1)/m<BR>\n" );
document.write( "Product of roots: x1 * x2 = c/a = (m + 1)/m\r<BR>\n" );
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document.write( "Let's manipulate these equations to eliminate m.\r<BR>\n" );
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document.write( "From the sum of roots:<BR>\n" );
document.write( "mx1 + mx2 = 2m - 2<BR>\n" );
document.write( "m(x1 + x2 - 2) = -2<BR>\n" );
document.write( "m = -2 / (x1 + x2 - 2)\r<BR>\n" );
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document.write( "From the product of roots:<BR>\n" );
document.write( "mx1x2 = m + 1<BR>\n" );
document.write( "m(x1x2 - 1) = 1<BR>\n" );
document.write( "m = 1 / (x1x2 - 1)\r<BR>\n" );
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document.write( "Equating the two expressions for m:<BR>\n" );
document.write( "-2 / (x1 + x2 - 2) = 1 / (x1x2 - 1)<BR>\n" );
document.write( "-2(x1x2 - 1) = x1 + x2 - 2<BR>\n" );
document.write( "-2x1x2 + 2 = x1 + x2 - 2<BR>\n" );
document.write( "x1 + x2 + 2x1x2 - 4 = 0\r<BR>\n" );
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document.write( "This is the relation independent of m.\r<BR>\n" );
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document.write( "**3. Double Root (x1 = x2 = X):**\r<BR>\n" );
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document.write( "Substitute x1 = x2 = X into the relation:<BR>\n" );
document.write( "X + X + 2X² - 4 = 0<BR>\n" );
document.write( "2X + 2X² - 4 = 0<BR>\n" );
document.write( "X² + X - 2 = 0<BR>\n" );
document.write( "(X + 2)(X - 1) = 0<BR>\n" );
document.write( "X = -2 or X = 1\r<BR>\n" );
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document.write( "**Case 1: X = -2**\r<BR>\n" );
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document.write( "Substitute X = -2 into the sum of roots:<BR>\n" );
document.write( "2X = 2(m - 1)/m<BR>\n" );
document.write( "2(-2) = 2(m - 1)/m<BR>\n" );
document.write( "-4m = 2m - 2<BR>\n" );
document.write( "-6m = -2<BR>\n" );
document.write( "m = 1/3\r<BR>\n" );
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document.write( "**Case 2: X = 1**\r<BR>\n" );
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document.write( "Substitute X = 1 into the sum of roots:<BR>\n" );
document.write( "2(1) = 2(m - 1)/m<BR>\n" );
document.write( "2m = 2m - 2<BR>\n" );
document.write( "0 = -2 (This is impossible)\r<BR>\n" );
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document.write( "Therefore, the only possible double root is X = -2, and the corresponding value of m is 1/3.\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "1.  Roots exist when m ≤ 1/3.<BR>\n" );
document.write( "2.  The relation independent of m is x1 + x2 + 2x1x2 - 4 = 0.<BR>\n" );
document.write( "3.  The possible double root is X = -2, and the corresponding value of m is 1/3.<BR>\n" );
document.write( " </SPAN>\n" );
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