(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "CalcMath"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "CalcMath"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "CalcMath"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "CalcMath"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "CalcMath"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "CalcMath"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "CalcMath"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "CalcMath"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l34, w351, h314, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "CalcMath"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "CalcMath"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Courier"; paletteColors = 256; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] ROOTS OF PARABOLAS :[font = subsubtitle; inactive; preserveAspect] By Ann Webbink :[font = subsubsection; inactive; Cclosed; preserveAspect; center; fontSize = 9; startGroup] Initialization :[font = input; initialization; preserveAspect; endGroup] *) Needs["Graphics`Colors`"] (* :[font = section; inactive; Cclosed; preserveAspect; startGroup] Getting Started :[font = smalltext; inactive; preserveAspect; endGroup] When studying the parabola, many students become confused about what is happening when the roots are no longer real roots. This program will attempt to show that there are always two roots to the quadratic equation, whether the roots be real, one root of multiplicity two, or two imaginary roots. :[font = section; inactive; Cclosed; preserveAspect; startGroup] The parabola with two real roots :[font = smalltext; inactive; preserveAspect] First let's look at a parabola with two real roots. :[font = input; preserveAspect; startGroup] Clear[f,x]; f[x_]:=x^2 -2x -8 Plot[f[x],{x,-5,5},AxesLabel->{"x","y"}, PlotRange->{-9,9}, PlotStyle->{Red}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 216; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0952381 0.309017 0.0343352 [ [(-4)] .11905 .30902 0 2 Msboxa [(-2)] .30952 .30902 0 2 Msboxa [(2)] .69048 .30902 0 2 Msboxa [(4)] .88095 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(-7.5)] .4875 .0515 1 0 Msboxa [(-5)] .4875 .13734 1 0 Msboxa [(-2.5)] .4875 .22318 1 0 Msboxa [(2.5)] .4875 .39486 1 0 Msboxa [(5)] .4875 .48069 1 0 Msboxa [(7.5)] .4875 .56653 1 0 Msboxa [(y)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .11905 .30902 m .11905 .31527 L s P [(-4)] .11905 .30902 0 2 Mshowa p .002 w .30952 .30902 m .30952 .31527 L s P [(-2)] .30952 .30902 0 2 Mshowa p .002 w .69048 .30902 m .69048 .31527 L s P [(2)] .69048 .30902 0 2 Mshowa p .002 w .88095 .30902 m .88095 .31527 L s P [(4)] .88095 .30902 0 2 Mshowa p .001 w .15714 .30902 m .15714 .31277 L s P p .001 w .19524 .30902 m .19524 .31277 L s P p .001 w .23333 .30902 m .23333 .31277 L s P p .001 w .27143 .30902 m .27143 .31277 L s P p .001 w .34762 .30902 m .34762 .31277 L s P p .001 w .38571 .30902 m .38571 .31277 L s P p .001 w .42381 .30902 m .42381 .31277 L s P p .001 w .4619 .30902 m .4619 .31277 L s P p .001 w .5381 .30902 m .5381 .31277 L s P p .001 w .57619 .30902 m .57619 .31277 L s P p .001 w .61429 .30902 m .61429 .31277 L s P p .001 w .65238 .30902 m .65238 .31277 L s P p .001 w .72857 .30902 m .72857 .31277 L s P p .001 w .76667 .30902 m .76667 .31277 L s P p .001 w .80476 .30902 m .80476 .31277 L s P p .001 w .84286 .30902 m .84286 .31277 L s P p .001 w .08095 .30902 m .08095 .31277 L s P p .001 w .04286 .30902 m .04286 .31277 L s P p .001 w .00476 .30902 m .00476 .31277 L s P p .001 w .91905 .30902 m .91905 .31277 L s P p .001 w .95714 .30902 m .95714 .31277 L s P p .001 w .99524 .30902 m .99524 .31277 L s P [(x)] 1.025 .30902 -1 0 Mshowa p .002 w 0 .30902 m 1 .30902 L s P p .002 w .5 .0515 m .50625 .0515 L s P [(-7.5)] .4875 .0515 1 0 Mshowa p .002 w .5 .13734 m .50625 .13734 L s P [(-5)] .4875 .13734 1 0 Mshowa p .002 w .5 .22318 m .50625 .22318 L s P [(-2.5)] .4875 .22318 1 0 Mshowa p .002 w .5 .39486 m .50625 .39486 L s P [(2.5)] .4875 .39486 1 0 Mshowa p .002 w .5 .48069 m .50625 .48069 L s P [(5)] .4875 .48069 1 0 Mshowa p .002 w .5 .56653 m .50625 .56653 L s P [(7.5)] .4875 .56653 1 0 Mshowa p .001 w .5 .06867 m .50375 .06867 L s P p .001 w .5 .08584 m .50375 .08584 L s P p .001 w .5 .10301 m .50375 .10301 L s P p .001 w .5 .12017 m .50375 .12017 L s P p .001 w .5 .15451 m .50375 .15451 L s P p .001 w .5 .17168 m .50375 .17168 L s P p .001 w .5 .18884 m .50375 .18884 L s P p .001 w .5 .20601 m .50375 .20601 L s P p .001 w .5 .24035 m .50375 .24035 L s P p .001 w .5 .25751 m .50375 .25751 L s P p .001 w .5 .27468 m .50375 .27468 L s P p .001 w .5 .29185 m .50375 .29185 L s P p .001 w .5 .32618 m .50375 .32618 L s P p .001 w .5 .34335 m .50375 .34335 L s P p .001 w .5 .36052 m .50375 .36052 L s P p .001 w .5 .37769 m .50375 .37769 L s P p .001 w .5 .41202 m .50375 .41202 L s P p .001 w .5 .42919 m .50375 .42919 L s P p .001 w .5 .44636 m .50375 .44636 L s P p .001 w .5 .46353 m .50375 .46353 L s P p .001 w .5 .49786 m .50375 .49786 L s P p .001 w .5 .51503 m .50375 .51503 L s P p .001 w .5 .5322 m .50375 .5322 L s P p .001 w .5 .54936 m .50375 .54936 L s P p .001 w .5 .03434 m .50375 .03434 L s P p .001 w .5 .01717 m .50375 .01717 L s P p .001 w .5 .5837 m .50375 .5837 L s P p .001 w .5 .60087 m .50375 .60087 L s P [(y)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p 1 0 0 r p .004 w s s s s s .19152 .61803 m .22222 .52671 L s .22222 .52671 m .2619 .42061 L .30159 .32642 L .34127 .24416 L .38095 .17382 L .42063 .1154 L .46032 .06891 L .48016 .05013 L .5 .03434 L .51984 .02152 L .53968 .01168 L .5496 .00788 L .55952 .00483 L .56448 .00358 L .56944 .00252 L .5744 .00164 L .57937 .00095 L .58185 .00068 L .58433 .00045 L .58681 .00027 L .58805 .0002 L .58929 .00013 L .59053 8e-05 L .59177 5e-05 L .59301 2e-05 L .59425 0 L .59549 0 L .59673 1e-05 L .59797 3e-05 L .59921 6e-05 L .60045 .0001 L .60169 .00016 L .60417 .0003 L .60665 .00049 L .60913 .00073 L .61409 .00134 L .61905 .00215 L .62897 .00431 L .63889 .00721 L .65873 .01526 L .67857 .02629 L .69841 .0403 L .7381 .07725 L .77778 .12613 L .81746 .18694 L .85714 .25966 L .89683 .34431 L .93651 .44087 L .97619 .54936 L s P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; endGroup] As we can see, the graph of the parabola crosses the x-axis in two places, thus giving us two real roots for this quadratic equation. Also notice that the sign of the function changes at these points since the graph crosses the x-axis at the points. :[font = section; inactive; Cclosed; preserveAspect; startGroup] The parabola with one root of multiplicity two :[font = smalltext; inactive; preserveAspect] Now let's examine what happens when a quadratic has only one root. When this happens what would you expect the graph of the parabola to look like? How many times should the graph cross the x-axis? :[font = input; preserveAspect; startGroup] Clear[f,x]; f[x_]:=x^2-2x+1 Plot[f[x],{x,-5,5},AxesLabel->{"x","y"}, PlotRange->{-5,5},AspectRatio->1,PlotStyle->{Red}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0952381 0.5 0.1 [ [(-4)] .11905 .5 0 2 Msboxa [(-2)] .30952 .5 0 2 Msboxa [(2)] .69048 .5 0 2 Msboxa [(4)] .88095 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 .1 1 0 Msboxa [(-2)] .4875 .3 1 0 Msboxa [(2)] .4875 .7 1 0 Msboxa [(4)] .4875 .9 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .11905 .5 m .11905 .50625 L s P [(-4)] .11905 .5 0 2 Mshowa p .002 w .30952 .5 m .30952 .50625 L s P [(-2)] .30952 .5 0 2 Mshowa p .002 w .69048 .5 m .69048 .50625 L s P [(2)] .69048 .5 0 2 Mshowa p .002 w .88095 .5 m .88095 .50625 L s P [(4)] .88095 .5 0 2 Mshowa p .001 w .15714 .5 m .15714 .50375 L s P p .001 w .19524 .5 m .19524 .50375 L s P p .001 w .23333 .5 m .23333 .50375 L s P p .001 w .27143 .5 m .27143 .50375 L s P p .001 w .34762 .5 m .34762 .50375 L s P p .001 w .38571 .5 m .38571 .50375 L s P p .001 w .42381 .5 m .42381 .50375 L s P p .001 w .4619 .5 m .4619 .50375 L s P p .001 w .5381 .5 m .5381 .50375 L s P p .001 w .57619 .5 m .57619 .50375 L s P p .001 w .61429 .5 m .61429 .50375 L s P p .001 w .65238 .5 m .65238 .50375 L s P p .001 w .72857 .5 m .72857 .50375 L s P p .001 w .76667 .5 m .76667 .50375 L s P p .001 w .80476 .5 m .80476 .50375 L s P p .001 w .84286 .5 m .84286 .50375 L s P p .001 w .08095 .5 m .08095 .50375 L s P p .001 w .04286 .5 m .04286 .50375 L s P p .001 w .00476 .5 m .00476 .50375 L s P p .001 w .91905 .5 m .91905 .50375 L s P p .001 w .95714 .5 m .95714 .50375 L s P p .001 w .99524 .5 m .99524 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 .1 m .50625 .1 L s P [(-4)] .4875 .1 1 0 Mshowa p .002 w .5 .3 m .50625 .3 L s P [(-2)] .4875 .3 1 0 Mshowa p .002 w .5 .7 m .50625 .7 L s P [(2)] .4875 .7 1 0 Mshowa p .002 w .5 .9 m .50625 .9 L s P [(4)] .4875 .9 1 0 Mshowa p .001 w .5 .14 m .50375 .14 L s P p .001 w .5 .18 m .50375 .18 L s P p .001 w .5 .22 m .50375 .22 L s P p .001 w .5 .26 m .50375 .26 L s P p .001 w .5 .34 m .50375 .34 L s P p .001 w .5 .38 m .50375 .38 L s P p .001 w .5 .42 m .50375 .42 L s P p .001 w .5 .46 m .50375 .46 L s P p .001 w .5 .54 m .50375 .54 L s P p .001 w .5 .58 m .50375 .58 L s P p .001 w .5 .62 m .50375 .62 L s P p .001 w .5 .66 m .50375 .66 L s P p .001 w .5 .74 m .50375 .74 L s P p .001 w .5 .78 m .50375 .78 L s P p .001 w .5 .82 m .50375 .82 L s P p .001 w .5 .86 m .50375 .86 L s P p .001 w .5 .06 m .50375 .06 L s P p .001 w .5 .02 m .50375 .02 L s P p .001 w .5 .94 m .50375 .94 L s P p .001 w .5 .98 m .50375 .98 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p 1 0 0 r p .004 w s s s s s s s s s s .38241 1 m .42063 .83611 L s .42063 .83611 m .46032 .70069 L .48016 .64601 L .5 .6 L .51984 .56267 L .53968 .53403 L .5496 .52296 L .55952 .51406 L .56448 .51043 L .56944 .50734 L .5744 .50479 L .57937 .50278 L .58185 .50198 L .58433 .50131 L .58681 .50078 L .58805 .50057 L .58929 .50039 L .59053 .50024 L .59177 .50013 L .59301 .50005 L .59425 .50001 L .59549 .5 L .59673 .50002 L .59797 .50008 L .59921 .50017 L .60045 .5003 L .60169 .50046 L .60417 .50088 L .60665 .50144 L .60913 .50213 L .61409 .50392 L .61905 .50625 L .62897 .51254 L .63889 .52101 L .65873 .54444 L .67857 .57656 L .69841 .61736 L .7381 .725 L .77778 .86736 L s .8075 1 m .77778 .86736 L s s s s s s P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; endGroup] Notice that this graph only touches the x-axis, and that the function does not change signs at this point. If we had factored this quadratic equation to solve for the roots we would have found the following equation: xÛ-2x +1 = 0. In factored form we would have (x - 1)(x - 1) = 0, giving us two identical roots at x = 1. This is called a root of multiplicity two since it occurs twice. :[font = section; inactive; Cclosed; preserveAspect; startGroup] A quadratic equation with two imaginary roots :[font = smalltext; inactive; preserveAspect] Now, what would you expect the graph of the quadratic that does not have any real roots to look like? Notice that in the two previous examples, we have changed only the value of the constant in the quadratic equation and have changed the nature of the roots of the equation. Let's try this one more time and examine the graph of the resulting parabola. :[font = input; preserveAspect; startGroup] Clear[f,x]; f[x_]:=x^2 +2x +3; Plot[f[x],{x,-5,5},AxesLabel->{"x","y"},PlotRange->{-5,5}, AspectRatio->1,PlotStyle->{Red}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0952381 0.5 0.1 [ [(-4)] .11905 .5 0 2 Msboxa [(-2)] .30952 .5 0 2 Msboxa [(2)] .69048 .5 0 2 Msboxa [(4)] .88095 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 .1 1 0 Msboxa [(-2)] .4875 .3 1 0 Msboxa [(2)] .4875 .7 1 0 Msboxa [(4)] .4875 .9 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .11905 .5 m .11905 .50625 L s P [(-4)] .11905 .5 0 2 Mshowa p .002 w .30952 .5 m .30952 .50625 L s P [(-2)] .30952 .5 0 2 Mshowa p .002 w .69048 .5 m .69048 .50625 L s P [(2)] .69048 .5 0 2 Mshowa p .002 w .88095 .5 m .88095 .50625 L s P [(4)] .88095 .5 0 2 Mshowa p .001 w .15714 .5 m .15714 .50375 L s P p .001 w .19524 .5 m .19524 .50375 L s P p .001 w .23333 .5 m .23333 .50375 L s P p .001 w .27143 .5 m .27143 .50375 L s P p .001 w .34762 .5 m .34762 .50375 L s P p .001 w .38571 .5 m .38571 .50375 L s P p .001 w .42381 .5 m .42381 .50375 L s P p .001 w .4619 .5 m .4619 .50375 L s P p .001 w .5381 .5 m .5381 .50375 L s P p .001 w .57619 .5 m .57619 .50375 L s P p .001 w .61429 .5 m .61429 .50375 L s P p .001 w .65238 .5 m .65238 .50375 L s P p .001 w .72857 .5 m .72857 .50375 L s P p .001 w .76667 .5 m .76667 .50375 L s P p .001 w .80476 .5 m .80476 .50375 L s P p .001 w .84286 .5 m .84286 .50375 L s P p .001 w .08095 .5 m .08095 .50375 L s P p .001 w .04286 .5 m .04286 .50375 L s P p .001 w .00476 .5 m .00476 .50375 L s P p .001 w .91905 .5 m .91905 .50375 L s P p .001 w .95714 .5 m .95714 .50375 L s P p .001 w .99524 .5 m .99524 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 .1 m .50625 .1 L s P [(-4)] .4875 .1 1 0 Mshowa p .002 w .5 .3 m .50625 .3 L s P [(-2)] .4875 .3 1 0 Mshowa p .002 w .5 .7 m .50625 .7 L s P [(2)] .4875 .7 1 0 Mshowa p .002 w .5 .9 m .50625 .9 L s P [(4)] .4875 .9 1 0 Mshowa p .001 w .5 .14 m .50375 .14 L s P p .001 w .5 .18 m .50375 .18 L s P p .001 w .5 .22 m .50375 .22 L s P p .001 w .5 .26 m .50375 .26 L s P p .001 w .5 .34 m .50375 .34 L s P p .001 w .5 .38 m .50375 .38 L s P p .001 w .5 .42 m .50375 .42 L s P p .001 w .5 .46 m .50375 .46 L s P p .001 w .5 .54 m .50375 .54 L s P p .001 w .5 .58 m .50375 .58 L s P p .001 w .5 .62 m .50375 .62 L s P p .001 w .5 .66 m .50375 .66 L s P p .001 w .5 .74 m .50375 .74 L s P p .001 w .5 .78 m .50375 .78 L s P p .001 w .5 .82 m .50375 .82 L s P p .001 w .5 .86 m .50375 .86 L s P p .001 w .5 .06 m .50375 .06 L s P p .001 w .5 .02 m .50375 .02 L s P p .001 w .5 .94 m .50375 .94 L s P p .001 w .5 .98 m .50375 .98 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p 1 0 0 r p .004 w s s s s s s .241 1 m .2619 .925 L s .2619 .925 m .28175 .86684 L .30159 .81736 L .32143 .77656 L .34127 .74444 L .35119 .73164 L .36111 .72101 L .37103 .71254 L .37599 .70913 L .38095 .70625 L .38591 .70392 L .38839 .70295 L .39087 .70213 L .39335 .70144 L .39583 .70088 L .39707 .70065 L .39831 .70046 L .39955 .7003 L .40079 .70017 L .40203 .70008 L .40327 .70002 L .40451 .7 L .40575 .70001 L .40699 .70005 L .40823 .70013 L .40947 .70024 L .41071 .70039 L .41319 .70078 L .41567 .70131 L .41815 .70198 L .42063 .70278 L .4256 .70479 L .43056 .70734 L .44048 .71406 L .4504 .72296 L .46032 .73403 L .48016 .76267 L .5 .8 L .53968 .90069 L s .56878 1 m .53968 .90069 L s s s s s s s s s s s s P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect] Did you notice that the graph no longer even touches the x-axis? Does that mean that there are suddenly no roots for this quadratic equation? Would that make any sense? We can find roots if we use the quadratic formula, but these roots would be imaginary. First, let's look at the roots when we let c vary from -2 to 4 in increments of one. :[font = input; preserveAspect; startGroup] Clear[f,x,c,comrt,sol,g,parabgraph ,i]; f[x_,c_]:= x^2 -2x +c; sol = Table[N[Solve[f[x,c]==0,x]],{c,-2,4}] :[font = output; output; inactive; preserveAspect; endGroup] {{{x -> -0.7320508075688772936}, {x -> 2.732050807568877294}}, {{x -> -0.4142135623730950488}, {x -> 2.414213562373095049}}, {{x -> 0}, {x -> 2.}}, {{x -> 1.}, {x -> 1.}}, {{x -> 1. - 1.*I}, {x -> 1. + 1.*I}}, {{x -> 1. - 1.414213562373095049*I}, {x -> 1. + 1.414213562373095049*I}}, {{x -> 1. - 1.732050807568877294*I}, {x -> 1. + 1.732050807568877294*I}}} ;[o] {{{x -> -0.732051}, {x -> 2.73205}}, {{x -> -0.414214}, {x -> 2.41421}}, {{x -> 0}, {x -> 2.}}, {{x -> 1.}, {x -> 1.}}, {{x -> 1. - 1. I}, {x -> 1. + 1. I}}, {{x -> 1. - 1.41421 I}, {x -> 1. + 1.41421 I}}, {{x -> 1. - 1.73205 I}, {x -> 1. + 1.73205 I}}} :[font = smalltext; inactive; preserveAspect] Notice that the first four roots are real, with the fourth root being a root of multiplicity two, or a double root. The last three roots are imaginary. To see what is happening, let's examine the graph of these successive quadratic equations. The graph to the right is the graph of the parabola on the real plane. The graph to the left is the graph of only the roots on the imaginary plane. :[font = input; preserveAspect; startGroup] comrt=Table[{Re[sol[[j,i,1,2]]],Im[sol[[j,i,1,2]]]}, {j,1,Length[sol]}, {i,1,2}]; g[i_]:=ListPlot[comrt[[i]], PlotRange->{{-5.0,5.0},{-5.0,5.0}}, AxesLabel->{"a","b"}, PlotStyle->{Blue,PointSize[0.05]}, AspectRatio->1,DisplayFunction -> Identity]; parabgraph := Plot[f[x,c],{x,-5,5},AxesLabel->{"x","y"}, PlotRange->{-5,5},AspectRatio->1,PlotStyle->{Red}, DisplayFunction->Identity]; Table[Show[GraphicsArray[{g[c+3],parabgraph}, DisplayFunction->$DisplayFunction]],{c,-2,4}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 167; startGroup; animationSpeed = 23; infiniteLoop; loopDistance = 1] %! %%Creator: Mathematica %%AspectRatio: .47619 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0113379 0.47619 [ [ 0 0 0 0 ] [ 1 .47619 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .47619 L 0 .47619 L closepath clip newpath p p % Start of sub-graphic p 0.0238095 0.0113379 0.477324 0.464853 MathSubStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.1 0.5 0.1 [ [(-4)] .1 .5 0 2 Msboxa [(-2)] .3 .5 0 2 Msboxa [(2)] .7 .5 0 2 Msboxa [(4)] .9 .5 0 2 Msboxa [(a)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 .1 1 0 Msboxa [(-2)] .4875 .3 1 0 Msboxa [(2)] .4875 .7 1 0 Msboxa [(4)] .4875 .9 1 0 Msboxa [(b)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash p p .002 w .1 .5 m .1 .50625 L s P [(-4)] .1 .5 0 2 Mshowa p .002 w .3 .5 m .3 .50625 L s P [(-2)] .3 .5 0 2 Mshowa p .002 w .7 .5 m .7 .50625 L s P [(2)] .7 .5 0 2 Mshowa p .002 w .9 .5 m .9 .50625 L s P [(4)] .9 .5 0 2 Mshowa p .001 w .14 .5 m .14 .50375 L s P p .001 w .18 .5 m .18 .50375 L s P p .001 w .22 .5 m .22 .50375 L s P p .001 w .26 .5 m .26 .50375 L s P p .001 w .34 .5 m .34 .50375 L s P p .001 w .38 .5 m .38 .50375 L s P p .001 w .42 .5 m .42 .50375 L s P p .001 w .46 .5 m .46 .50375 L s P p .001 w .54 .5 m .54 .50375 L s P p .001 w .58 .5 m .58 .50375 L s P p .001 w .62 .5 m .62 .50375 L s P p .001 w .66 .5 m .66 .50375 L s P p .001 w .74 .5 m .74 .50375 L s P p .001 w .78 .5 m .78 .50375 L s P p .001 w .82 .5 m .82 .50375 L s P p .001 w .86 .5 m .86 .50375 L s P p .001 w .06 .5 m .06 .50375 L s P p .001 w .02 .5 m .02 .50375 L s P p .001 w .94 .5 m .94 .50375 L s P p .001 w .98 .5 m .98 .50375 L s P [(a)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 .1 m .50625 .1 L s P [(-4)] .4875 .1 1 0 Mshowa p .002 w .5 .3 m .50625 .3 L s P [(-2)] .4875 .3 1 0 Mshowa p .002 w .5 .7 m .50625 .7 L s P [(2)] .4875 .7 1 0 Mshowa p .002 w .5 .9 m .50625 .9 L s P [(4)] .4875 .9 1 0 Mshowa p .001 w .5 .14 m .50375 .14 L s P p .001 w .5 .18 m .50375 .18 L s P p .001 w .5 .22 m .50375 .22 L s P p .001 w .5 .26 m .50375 .26 L s P p .001 w .5 .34 m .50375 .34 L s P p .001 w .5 .38 m .50375 .38 L s P p .001 w .5 .42 m .50375 .42 L s P p .001 w .5 .46 m .50375 .46 L s P p .001 w .5 .54 m .50375 .54 L s P p .001 w .5 .58 m .50375 .58 L s P p .001 w .5 .62 m .50375 .62 L s P p .001 w .5 .66 m .50375 .66 L s P p .001 w .5 .74 m .50375 .74 L s P p .001 w .5 .78 m .50375 .78 L s P p .001 w .5 .82 m .50375 .82 L s P p .001 w .5 .86 m .50375 .86 L s P p .001 w .5 .06 m .50375 .06 L s P p .001 w .5 .02 m .50375 .02 L s P p .001 w .5 .94 m .50375 .94 L s P p .001 w .5 .98 m .50375 .98 L s P [(b)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p 0 0 1 r p .05 w .42679 .5 Mdot .77321 .5 Mdot P P MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.522676 0.0113379 0.97619 0.464853 MathSubStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0952381 0.5 0.1 [ [(-4)] .11905 .5 0 2 Msboxa [(-2)] .30952 .5 0 2 Msboxa [(2)] .69048 .5 0 2 Msboxa [(4)] .88095 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 .1 1 0 Msboxa [(-2)] .4875 .3 1 0 Msboxa [(2)] .4875 .7 1 0 Msboxa [(4)] .4875 .9 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash p p .002 w .11905 .5 m .11905 .50625 L s P [(-4)] .11905 .5 0 2 Mshowa p .002 w .30952 .5 m .30952 .50625 L s P [(-2)] .30952 .5 0 2 Mshowa p .002 w .69048 .5 m .69048 .50625 L s P [(2)] .69048 .5 0 2 Mshowa p .002 w .88095 .5 m .88095 .50625 L s P [(4)] .88095 .5 0 2 Mshowa p .001 w .15714 .5 m .15714 .50375 L s P p .001 w .19524 .5 m .19524 .50375 L s P p .001 w .23333 .5 m .23333 .50375 L s P p .001 w .27143 .5 m .27143 .50375 L s P p .001 w .34762 .5 m .34762 .50375 L s P p .001 w .38571 .5 m .38571 .50375 L s P p .001 w .42381 .5 m .42381 .50375 L s P p .001 w .4619 .5 m .4619 .50375 L s P p .001 w .5381 .5 m .5381 .50375 L s P p .001 w .57619 .5 m .57619 .50375 L s P p .001 w .61429 .5 m .61429 .50375 L s P p .001 w .65238 .5 m .65238 .50375 L s P p .001 w .72857 .5 m .72857 .50375 L s P p .001 w .76667 .5 m .76667 .50375 L s P p .001 w .80476 .5 m .80476 .50375 L s P p .001 w .84286 .5 m .84286 .50375 L s P p .001 w .08095 .5 m .08095 .50375 L s P p .001 w .04286 .5 m .04286 .50375 L s P p .001 w .00476 .5 m .00476 .50375 L s P p .001 w .91905 .5 m .91905 .50375 L s P p .001 w .95714 .5 m .95714 .50375 L s P p .001 w .99524 .5 m .99524 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 .1 m .50625 .1 L s P [(-4)] .4875 .1 1 0 Mshowa p .002 w .5 .3 m .50625 .3 L s P [(-2)] .4875 .3 1 0 Mshowa p .002 w .5 .7 m .50625 .7 L s P [(2)] .4875 .7 1 0 Mshowa p .002 w .5 .9 m .50625 .9 L s P [(4)] .4875 .9 1 0 Mshowa p .001 w .5 .14 m .50375 .14 L s P p .001 w .5 .18 m .50375 .18 L s P p .001 w .5 .22 m .50375 .22 L s P p .001 w .5 .26 m .50375 .26 L s P p .001 w .5 .34 m .50375 .34 L s P p .001 w .5 .38 m .50375 .38 L s P p .001 w .5 .42 m .50375 .42 L s P p .001 w .5 .46 m .50375 .46 L s P p .001 w .5 .54 m .50375 .54 L s P p .001 w .5 .58 m .50375 .58 L s P p .001 w .5 .62 m .50375 .62 L s P p .001 w .5 .66 m .50375 .66 L s P p .001 w .5 .74 m .50375 .74 L s P p .001 w .5 .78 m .50375 .78 L s P p .001 w .5 .82 m .50375 .82 L s P p .001 w .5 .86 m .50375 .86 L s P p .001 w .5 .06 m .50375 .06 L s P p .001 w .5 .02 m .50375 .02 L s P p .001 w .5 .94 m .50375 .94 L s P p .001 w .5 .98 m .50375 .98 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p 1 0 0 r p .004 w s s s s s s s s .32655 1 m .34127 .91111 L s .34127 .91111 m .38095 .70625 L .42063 .53611 L .46032 .40069 L .48016 .34601 L .5 .3 L .51984 .26267 L .53968 .23403 L .5496 .22296 L .55952 .21406 L .56448 .21043 L .56944 .20734 L .5744 .20479 L .57937 .20278 L .58185 .20198 L .58433 .20131 L .58681 .20078 L .58805 .20057 L .58929 .20039 L .59053 .20024 L .59177 .20013 L .59301 .20005 L .59425 .20001 L .59549 .2 L .59673 .20002 L .59797 .20008 L .59921 .20017 L .60045 .2003 L .60169 .20046 L .60417 .20088 L .60665 .20144 L .60913 .20213 L .61409 .20392 L .61905 .20625 L .62897 .21254 L .63889 .22101 L .65873 .24444 L .67857 .27656 L .69841 .31736 L .7381 .425 L .77778 .56736 L .81746 .74444 L .85714 .95625 L s .86419 1 m .85714 .95625 L s s s s P P MathSubEnd P % End of sub-graphic P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 167; animationSpeed = 23] %! %%Creator: Mathematica %%AspectRatio: .47619 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0113379 0.47619 [ [ 0 0 0 0 ] [ 1 .47619 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .47619 L 0 .47619 L 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.80213 L .61409 .80392 L .61905 .80625 L .62897 .81254 L .63889 .82101 L .65873 .84444 L .67857 .87656 L .69841 .91736 L s .72888 1 m .69841 .91736 L s s s s s s s s P P MathSubEnd P % End of sub-graphic P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; endGroup; endGroup] Did you notice that when the graph of the parabola "lifted" off the x=axis, the roots on the imaginary plane left the real axis and switched to the imaginary axis? This shows that we will always have two roots to a quadratic equation, whether they be two real roots, one root of multiplicity two, or two imaginary roots. ^*)