document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=15351>Question 15351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Write an equation in the form y=ax^2+bx+c for the quadratic function whose graph passes through (8,0),(0,8) and (-2,0).<BR>\n" );
document.write( "2. Find the roots of x^2 + (k^2+1 over k)x+1=0.<BR>\n" );
document.write( "3. If (2 over x - x over 2)^2=o,evaluate x^6.<BR>\n" );
document.write( "4. Find all values of k that ensure that the roots are real for x-k(x-1)(x-2)=o.<BR>\n" );
document.write( "5. Find all possible values of k so that 3x^2 + kx +5 can be factored as the product of two binomial factors with integer cefficients.<BR>\n" );
document.write( "6. Show that there are nine pairs of positive integers (m,n) such that m^2+3mn+2n^2-10m-20n=0.<BR>\n" );
document.write( "7. The difference in the length of the hypotenuse of triangleABC and the length of the hypotenuse of triangleXYZ is 3. Hypotenuse AB=x,hypotenuse XY=square root x-1 and AB>XY. Determine the length of each hypotenuse.<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 #7644 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=7644\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  You posted so many questions without showing any work, <BR>\n" );
document.write( " I don't quite feel happy to solve for you.\r<BR>\n" );
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document.write( " 1.	Write an equation in the form y=ax^2+bx+c for the quadratic function whose graph passes through (8,0),(0,8) and (-2,0).<BR>\n" );
document.write( "Sol: Clearly, c = 8, then solve <BR>\n" );
document.write( "0 = 64a + 8b + 8 (or 8a + b = -1) and<BR>\n" );
document.write( "0 = 4a – 2b (or b = 2a) <BR>\n" );
document.write( "     for a and b = ???\r<BR>\n" );
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document.write( "2.Find the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+%2B+%28k%5E2%2B1%29%2Fk+%2A+x%2B1+ ALIGN=MIDDLE  ALT=\"+x%5E2+%2B+%28k%5E2%2B1%29%2Fk+%2A+x%2B1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=0.<BR>\n" );
document.write( "Sol: Use the quadratic formula directly.\r<BR>\n" );
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document.write( "3. If (2 /x – x/ 2)^2 = 0,evaluate x^6.<BR>\n" );
document.write( "Sol  2 /x – x/ 2  = 0, so (4-x^2) /2x= 0, <BR>\n" );
document.write( "or x = (+/-) 2, Now x^6 = ???\r<BR>\n" );
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document.write( "4. Find all values of k that ensure that the roots are real for x-k(x-1)(x-2)=o.<BR>\n" );
document.write( "Sol: Convert it to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-k+x%5E2+%2B+%283+k%2B1%29+x+-2k+ ALIGN=MIDDLE  ALT=\"-k+x%5E2+%2B+%283+k%2B1%29+x+-2k+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0 or<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+x%5E2+-+%283+k%2B1%29+x+%2B+2k ALIGN=MIDDLE  ALT=\"k+x%5E2+-+%283+k%2B1%29+x+%2B+2k\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0<BR>\n" );
document.write( "Since if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+-+4ac+=+%283+k%2B1%29%5E2+-+8k%5E2+%3E=0 ALIGN=MIDDLE  ALT=\"+b%5E2+-+4ac+=+%283+k%2B1%29%5E2+-+8k%5E2+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <BR>\n" );
document.write( "the given equation has only real roots<BR>\n" );
document.write( "Implying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2+%2B+6k+%2B+1+%3E=+0 ALIGN=MIDDLE  ALT=\"k%5E2+%2B+6k+%2B+1+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B3%29%5E2+%3E=8 ALIGN=MIDDLE  ALT=\"%28k%2B3%29%5E2+%3E=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Hence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+%3E=+-3%2B+2+sqrt%282%29 ALIGN=MIDDLE  ALT=\"k+%3E=+-3%2B+2+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+k+%3C=+-3+-+2+sqrt%282%29 ALIGN=MIDDLE  ALT=\"+k+%3C=+-3+-+2+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "5. Find all possible values of k so that 3x^2 + kx +5 can be factored as the product of two binomial factors with integer coefficients.<BR>\n" );
document.write( "Sol: 3x^2 + kx +5 = (3x+1)(x+5) or (3x-1)(x-5) or (3x+5)(x+1) or (3x-5)(x-1).<BR>\n" );
document.write( "Hence, k = ????\r<BR>\n" );
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document.write( "6. Show that there are nine pairs of positive integers (m,n) such that <BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++m%5E2%2B3mn%2B2n%5E2-10m-20n=0. ALIGN=MIDDLE  ALT=\"++m%5E2%2B3mn%2B2n%5E2-10m-20n=0.\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Sol: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2B3mn%2B2n%5E2-10m-20n ALIGN=MIDDLE  ALT=\"m%5E2%2B3mn%2B2n%5E2-10m-20n\"   DISABLED_style= \"max-width:90%; height:auto;\" > = (m + n)(m + 2n) –10(m+2n)<BR>\n" );
document.write( " = (m + 2n) (m+n-10). = 0.<BR>\n" );
document.write( "Hence, m + 2n = 0 (no positive sol.)<BR>\n" );
document.write( "Or m + n = 10, so there are 9 solutions <BR>\n" );
document.write( "{(1,9)(2,8),(3,7),(4,6),(5,5),(6,4),(3,7),(2,8)(1,9)}\r<BR>\n" );
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document.write( "7. The difference in the length of the hypotenuse of triangle ABC and the length of the hypotenuse of triangle XYZ is 3. Hypotenuse AB=x, hypotenuse XY= square root x-1 and AB>XY. Determine the length of each hypotenuse.<BR>\n" );
document.write( "Sol: Solve the equation x – sqrt(x-1) = 3 or x-3 = sqrt(x-1) <BR>\n" );
document.write( " So, (x-3)^2 = x –1 to get x^2-7x +10 = 0 and x = 2 or –5.(invalid)\r<BR>\n" );
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document.write( " Bad notation:1. Never use \"over\" in fraction again. <BR>\n" );
document.write( "              2. Square root x-1 means ????\r<BR>\n" );
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document.write( " You should work very hard.\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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