document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=880222>Question 880222</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1) Find the range of m such that the equation |x^2-3x+2|=mx has 4 distinct real solutions a,b,c,d<BR>\n" );
document.write( "(2) Express the value of s(m) = 1/a^2 + 1/b^2 + 1/c^2 + 1/d^2<BR>\n" );
document.write( "(3) when m varies as in (1) find the range of s(m) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #531365 by </B><A HREF=/tutors/aboutme.mpl?userid=DrBeeee><B>DrBeeee(684)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=DrBeeee><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=531365\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given;<BR>\n" );
document.write( "(1) y = |x^2-3x+2| and<BR>\n" );
document.write( "(2) y = mx<BR>\n" );
document.write( "The parabola opens upward with symmetry around<BR>\n" );
document.write( "(3) x = -b/(2*a) or<BR>\n" );
document.write( "(4) x = 3/2<BR>\n" );
document.write( "At x = 3/2, (1) gives<BR>\n" );
document.write( "(3) y = |1.5^@ -3*1.5 +2| or<BR>\n" );
document.write( "(4) y = |2.25 - 4.5 +2| or<BR>\n" );
document.write( "(5) y = |-.25| or<BR>\n" );
document.write( "(6) y = .25<BR>\n" );
document.write( "In order to have 4 distinct crossings (solutions) of (1) and (2) m must be greater than zero and less than <BR>\n" );
document.write( "(7) .25 = m*1.5 or<BR>\n" );
document.write( "(8) m = 1/6 giving the range of m as<BR>\n" );
document.write( "(9) 0 < m < 1/6<BR>\n" );
document.write( "When m = 0 we get<BR>\n" );
document.write( "(10) {a,b,c,d} = {1,1,2,2} the point on the x axis where the parabola of (1) crosses.<BR>\n" );
document.write( "To get the set for m = 1/6 set (1) = (2) at m = 1/6<BR>\n" );
document.write( "(11) x^2-3x+2 = x/6<BR>\n" );
document.write( "Use the quadratic equation to find<BR>\n" );
document.write( "(12) x = {0.0326,0.8840)<BR>\n" );
document.write( "These are the values of a and d when m=1/6<BR>\n" );
document.write( "The values of b and c are 1/4 when m = 1/6<BR>\n" );
document.write( "put tyhe sets into the expression for s(m) and get<BR>\n" );
document.write( "(13) s(0) = 2.5 and<BR>\n" );
document.write( "(14) s(1/6) = 939.507<BR>\n" );
document.write( "Sorry, I made a big error in the above solution. It's a bit harder than I gave you above. Here's the right answer.<BR>\n" );
document.write( "My error is in the selection of m = 1/6. The value of m can be slightly greater than 1/6. It turns out to be approximately 0.1715 vs 0.1666. <BR>\n" );
document.write( "The straight line mx can increase until it just touches the (inverted) parabola,<BR>\n" );
document.write( "(15) y =  -x^2+3x-2 between 1<x<2.<BR>\n" );
document.write( "To get m we set the derivitive of (15) equal to the slope of mx and get<BR>\n" );
document.write( "(16) -2x + 3 = m or<BR>\n" );
document.write( "(17) x = (3-m)/2<BR>\n" );
document.write( "Now substitute (17) into (15) to get (after some algebra) <BR>\n" );
document.write( "(18) y = (1-m^2)/4<BR>\n" );
document.write( "Note that when m=0, (18) gives us y = 1/4, the maximum value of y where the slope is zero.<BR>\n" );
document.write( "Now to get the value of m we need to set (18) equal to mx, where x is given by (17), <BR>\n" );
document.write( "(19) 1-m^2)/4 = m*(3-m)/2<BR>\n" );
document.write( "Simplify (19) to get<BR>\n" );
document.write( "(20) m^2 - 6m + 1 = 0<BR>\n" );
document.write( "Use the quadratic equation to find<BR>\n" );
document.write( "(21) m = 3 +- sqrt(2)<BR>\n" );
document.write( "Chose the negative to get<BR>\n" );
document.write( "(22) m = 3 - sqrt(2) or<BR>\n" );
document.write( "(23) m = 0.1715 which slightly greater than 1/6.<BR>\n" );
document.write( "Now that we have m we can the range of m as<BR>\n" );
document.write( "(24) 0<m<(3-sqrt(2))<BR>\n" );
document.write( "We can get b and c, equal at tangent point<BR>\n" );
document.write( "(25) (x,y) = ((3-m)/2,(1-m^2)/4) or<BR>\n" );
document.write( "(26) b = c = (3-m)/2 or<BR>\n" );
document.write( "(27) b = c = (3-(3 - sqrt(2))/2 or<BR>\n" );
document.write( "(28) b = c = sqrt(2)/2 or<BR>\n" );
document.write( "(29) b = c = 1/sqrt(2)<BR>\n" );
document.write( "Now to get a and d we need to set the parabola of (1), without the absolute value sign, because we are outside 1<x<2, equal to mx and get<BR>\n" );
document.write( "(30) x^2-3x+2 = (3-sqrt(2))*x, and solve for x using the quadratic equation to get<BR>\n" );
document.write( "(31) x = (3-sqrt(2) +- sqrt(9-6*sqrt(2)) or<BR>\n" );
document.write( "(32) x = {0.868341,2.3032}<BR>\n" );
document.write( "The value in (32) correspond to a and d respective.<BR>\n" );
document.write( "Using the values of {a,b,c,d} at m = 0.1715 we get<BR>\n" );
document.write( "(33) s(0.1716) = 35.485\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );