document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209301>Question 1209301</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What real value of t produces the smallest value of the quadratic t^2 -9t - 36 + 8t^2 + 55t + 41? </SPAN>\n" );
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document.write( "Answer: <font color=red>-23/9</font><BR>\n" );
document.write( "This is approximately equal to -2.55556 where the 5's go on forever but we have to round at some point.\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "I'll use x in place of t.\r<BR>\n" );
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document.write( "x^2 -9x - 36 + 8x^2 + 55x + 41<BR>\n" );
document.write( "= (x^2 + 8x^2) + (-9x + 55x) + ( -36 + 41)<BR>\n" );
document.write( "= 9x^2 + 46x + 5\r<BR>\n" );
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document.write( "The vertex of y = ax^2+bx+c is located at (h,k) where h = -b/(2a)<BR>\n" );
document.write( "y = 9x^2 + 46x + 5 has the coefficients a = 9, b = 46, c = 5\r<BR>\n" );
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document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -46/(2*9)<BR>\n" );
document.write( "h = -46/18<BR>\n" );
document.write( "h = (-2*23)/(2*9)<BR>\n" );
document.write( "h = <font color=red>-23/9</font><BR>\n" );
document.write( "This is the x coordinate of the vertex. <BR>\n" );
document.write( "This x value will make 9x^2 + 46x + 5 as small as possible.<BR>\n" );
document.write( "Note that a = 9 is positive, so the parabola opens upward, which places the vertex at the lowest point.\r<BR>\n" );
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document.write( "Another approach would be to use the quadratic formula to solve 9x^2 + 46x + 5 = 0 to get the roots x = -5 and x = -1/9<BR>\n" );
document.write( "I'll leave the scratch work for the student to do.<BR>\n" );
document.write( "Or you could factor 9x^2 + 46x + 5 into (x+5)(9x+1) to be able to see the roots easily. \r<BR>\n" );
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document.write( "x = -5 and x = -1/9 are where the parabola crosses the x axis.<BR>\n" );
document.write( "The midpoint of the x intercepts is the x coordinate of the vertex.<BR>\n" );
document.write( "This is due to the parabola's mirror symmetry about the center line. \r<BR>\n" );
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document.write( "p,q are the roots<BR>\n" );
document.write( "h = x coordinate of the vertex (h,k)<BR>\n" );
document.write( "h = (1/2)*(p+q)<BR>\n" );
document.write( "h = (1/2)*( -5 + (-1/9) )<BR>\n" );
document.write( "h = (1/2)*( -45/9 + (-1/9) )<BR>\n" );
document.write( "h = (1/2)*(-46/9)<BR>\n" );
document.write( "h = (-46)/(2*9)<BR>\n" );
document.write( "h = (-2*23)/(2*9)<BR>\n" );
document.write( "h = <font color=red>-23/9</font><BR>\n" );
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