00010001010015QuadraticEquation.ACT0002020015EquationWithRoots.EAC01000000b08dɸDkäûxÙ[Quadratic Equations[With Their Roots\Authoráà\PMun Chou, Fong Mathsroof Consultancy Malaysia. e-mail: mcfong@mathsroof.comÿ[[ OBJECTIVE[I. To construct quadratic[equations with its roots;[[II. To learn that many[quadratic equations have[the same roots and then[discover their general[form.[[ Example 1:[The roots of a quadratic[equation are given as -3 [and 2.4. The task is to[construct the suitable[quadratic equation with [these given roots.[[ Solution:[One way is through the[expansion of (í¸-a)(í¸-b)[where a and b are the [roots of the equation.[Tap Expansion strip to [ see how...[\Expansion StripËßlR clear_a_zRdoneR(í¸-(-3))(í¸-2.4)Rx+3î’x-125Rexpand(x+3î’x-125)Rx2+3î’x5-365RÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁ aseq_histbNewFolde systemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[With given real roots of[-3 and 2.4, we obtained[the quadratic equation as[x2+3î’x5-365=0.[[Exploration 1:[An interesting question to[ask now: Is the equation[obtained an unique one?[[To answer the question,[lets draw the graph of[y1=x2+3î’x5-365 and its[ variations.[- Tap on GraphStrip 1.[- Draw y1 and observe [ that the curve passes [ through -3 and 2.4 [ on x-axis.[- Now select function y2[ and y3. Draw all three[ y1, y2 and y3 on the[ same axes. [- Tap [Analysis]î[G-Solve][ î [Root] to find the [ roots of all three y1,[ y2 and y3. [\ GraphStrip 1ÐʤGraph2D †Graph3D¤ †LISTSYS°4N†Modify ä$ †STATCALC N†STATSYS \N†Sequencel, †Sheet˜| †Sheet3D| †SolveEq†SolveLwr” †SolveUpr  †StupFLG1¬(†StupListÔD †StupPict$ †ViewWind< †summaryXl †y1Ä$H†y2è(H†y3$H†ä4 †ä@ †äL †äX †äd †äp †ä| †äˆ †ä” †ä  †ä¬ †ä¸ †äÄ †äÐ †äÜ †äè †ä ô †ä! †ä" †ä# †ä$$ †ä%0 †ä&< †ä'H †ä(T †ä)` †ä*l †ä+x †ä,„ †ä- †ä.œ †ä/¨¤ †ä0L †ä1X †ä2d †ä3p †ä4| †ä5ˆ †äE” †äF  †äH¬ †äI¸ †äJÄ †äKÐ †äLÜ †äMè †äNô †äO †äP †äQ †äR$ †äS0 †äT< †ä]H †ä^L †ä_P †ä`T †äaX †äb\ †ä”` †ä•l †äÍx †ä΄ †äÐ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁÝÁ aR6,-4.477bT systemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁsystemä]listsystemä^]=systemÝÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $  $(4<HLX\hp|„” ¤°´ÀÄÐØäèôø$(4<HP\`lx„ˆ”œxTðxý-7.7*x/p``-3/10x/™Yp7.7pfíá(x)ðxý-14.8x/H`-Tðxý0R+Tðxý16`fíâ(x)ðxý2R+Tðxý2Q+Tðxý2Tf(x))ðxý47.47x/tpî–ðxý-729/100-36)`î—ðxý56.7100g 3xx^2+3*x/5-36/5 3x2*(x^2+3*x/5-36/5) 3x-(x^2+3*x/5-36/5)`P`P`p`p ™uRcx`d™G6„"(1…0qy`#Y‡uYƒ ˜000000 $0<HT`l```` ™…`BRcy ™T[[(1) Clearly the three[curves pass x=-3 and[x=2.4. What does this[imply?[[(2) State 2 relationships[between y2, y3 and y1.[[Now try the next activity[before answering (3),(4)[and (5).[ GRAPH MODIFY[- Tap on GraphStrip 2.[- Select y1 and graph it[- Tap on [Analysis] î[ [Modify].[When equation of the[curve appears in the[graph message box,[select only the 'A' at[front but not the whole[ equation.[- Now tap on the right[ and left graph arrow[ of î† îˆ to modify A.[\ GraphStrip 2ÐÊlGraph2Dh †Graph3D| †LISTSYSˆ4N†Modify ¼$ †STATCALC àN†STATSYS è\N†SequenceD, †Sheetp| †Sheet3Dì| †SolveEqh†SolveLwrl †SolveUprx †StupFLG1„(†StupList¬D †StupPictð$ †ViewWind †y10(H†y2X$H†y3|$H†ä  †ä¬ †ä¸ †äÄ †äÐ †äÜ †äè †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä ` †ä!l †ä"x †ä#„ †ä$ †ä%œ †ä&¨ †ä'´ †ä(À †ä)Ì †ä*Ø †ä+ä †ä,ð †ä-ü †ä. †ä0 †ä1 †ä2, †ä38 †ä4D †ä5P †äE\ †äFh †äHt †äI€ †äJŒ †äK˜ †äL¤ †äM° †äN¼ †äOÈ †äPÔ †äQà †äRì †äSø †äT †ä] †ä^ †ä_ †ä` †äa  †äb$ †ä”( †ä•4 †äÍ@ †äÎL †äÐX †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á aseq_histbNewFolde systemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $  3xA*(x^2+3*x/5-36/5) 3xx^2+3*x/5-36/5-A 3xA*x^2+3*x/5-36/5`P`P`p`p ™`(1…0qy`#Y‡uYƒ ˜Y™)`` ™R[Watch as the curve is[transformed according to[ values of A. [[Repeat GRAPH MODIFY but[first with y2 then follow[by y3.[[(3) Describe what you[may have observed in[the exercise of modifying[A in y1, y2 and y3.[[(4) State 2 examples of[quadratic equations which[have roots of -3 and[2.4.[[(5) Which equation below[best describes quadratic[equations with roots of[ -3 and 2.4?[a) x2+3î’x5-365=A, [b) Aî’x2+3î’x5-365=0, or[c) Ax2+3î’x5-365=0[[###Explore 1 Notes###[(1)The graphs imply that[these equations have -3[and 2.4 as their roots,[ and that x2+3î’x5-365=0[is not an unique answer.[[(2)The relationships are:[(i)y2=2y1 and (ii)y3=-y1 [[(3) One would observe[that the roots of the[equation remain the same[as -3 and 2.4 while A of[y1 is being modified.[ Modify A of y2 shifts[the curve vertically while[modifying A of y3 shrinks[ the curve.[[(4),(5)[From our modify exercise[in (3), general form of[ quadratic equations with[ roots of -3 and 2.4 is[ obviously[ A(x2+3î’x5-365)=0,[where A is real,[#######################[[ Exercise 1:[Use similar approach to[show that many quadratic[equations may also be[formed when the roots[are repeated and real, [such as -2 and -2.[\Use hereËß|R clear_a_zRdoneR(í¸-(-2))(í¸-(-2))Rx+22RÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á aseq_histbNewFolde systemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O\Use hereÐÊÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á aseq_histbNewFolde systemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[[The approach we used in[Exploration 1 is not very[effective if the equations[we are discussing have[imaginary roots.[[Exploration 2:[To understand why, we[graph y1=x2+4î’x+5 and[its variations of y2 and[y3 in GraphStrip 3 to[ illustrate.[[###Explore 2 Notes###[ - Note that x2 +4î’x+5=0[has imaginary roots,since[its discriminant is [ 42 -4î1î5=-4[which is less than 0.[#######################[\ GraphStrip 3ÐÊ´Graph2D| †Graph3D †LISTSYSœ4N†Modify Ð$ †STATCALC ôN†STATSYS ü\N†SequenceX, †Sheet„| †Sheet3D| †SolveEq|†SolveLwr€ †SolveUprŒ †StupFLG1˜(†StupListÀD †StupPict$ †ViewWind( †y1DH†y2` H†y3€$H†ä¤ †ä° †ä¼ †äÈ †äÔ †äà †äì †äø †ä †ä †ä †ä( †ä4 †ä@ †äL †äX †ä d †ä!p †ä"| †ä#ˆ †ä$” †ä%  †ä&¬ †ä'¸ †ä(Ä †ä)Ð †ä*Ü †ä+è †ä,ô †ä- †ä. †ä/D †ä0\ †ä1h †ä2t †ä3€ †ä4Œ †ä5˜ †äE¤ †äF° †äH¼ †äIÈ †äJÔ †äKà †äLì †äMø †äN †äO †äP †äQ( †äR4 †äS@ †äTL †ä]X †ä^\ †ä_` †ä`d †äah †äbl †ä”p †ä•| †ä͈ †äΔ †äР †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á aseq_histbNewFolde systemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $   3xx^2+4*x+5 3x2*(x^2+4*x+5) 3x-3*(x^2+4*x+5)`P`P`àï`àïàï8–8– ™àï`àïàïàï(1…0qy`àï#Y‡uYƒàï ˜ $0<HT`lx„œ¨´ÀÌØä`p@`` €`p@`P`P``&1W‰G7 ™S[The graphs show that [these three curves do[not intersect with x-axis[Hence we would not be[able to graphically show[they meet at some points[ on x-axis. [[We can however employ[this strategy:[Show that the solution[of B(x2+4î’x+ 5)=0 always[produce the same exact[solution for many non-[zero real values of B.[ [- Tap on Explore Strip.[- An initial value of 1 is[ assigned to B.[- Try assigning different[ real values to B.[ Remember to tap [EXE][ after each change.[\Explore Strip Ë ßhR clear_a_zRdoneR1 îBR1RBî(í¸^2+4í¸+5)=0îeq1Rx2+4î’x+5=0R solve(eq1)Rx=-2-î,x=-2+îRÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á aseq_histbNewFolde systemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[Note that for any real[values of B assigned,[the roots are always[-2-î and -2+ î.(except[when 0 is assigned to B)[[This result is suffice to[imply that infinitely many[ quadratic equations have[imaginary roots -2-î and[-2+î.[ \About îá à¬î is a symbol denoting îP(-1). It is important in the discussion on Complex Number. The general form of complex number is a+îb, where a, b are real. ÿ[[eActA èUxB èUxeq Eeq1LLF 0à ;xxeq244F 1à  xeq3((F 1à x€eq5S)/(02000eFindMaxMin.EAC01000000c2a5Éøúy=-3x^(2)+2î’x+1ü÷Ι™™™™™`g’E(0 `g’E(0 vrALCuxyÈ@  -3îx^2+2îx+1¸Dkäûx£[Finding Max And Min[ With ROOTS\Authoráà\PMun Chou, Fong Mathsroof Consultancy Malaysia. e-mail: mcfong@mathsroof.comÿ[[ OBJECTIVE[To algebraically determine[the maximum or minimum[point of quadratic curves[using roots of the[corresponding quadratic[ equation.[[Exploration 1:[We begin by working on[this exercise to discover[a relationship between[the  curve of Aí¸2+bí¸+c[and  coefficient of í¸2,[which is A. Now[- Tap on Geometry Strip.[- Replace the coefficient[ of í¸2 in Geometry Link[ with -3,,2,-1 and 4.[ Remember to tap [EXE][ after each replacement[]\Geometry StripÎ[[Try describe what you[observe, then tap to see[notes...\Exploration Notesá àWe may conclude from this exercise that: - If coefficient of í¸íò is positive, the curve of aí¸íò+bí¸+c is a minimum curve/has minimum point, On the other hand, - if coefficient of í¸íò is negative, then the curve is a maximum curve/has maximum point. ÿ[[ Example 1:[The task here is to find[the minimum or maximum[point of y=í¸2+6í¸+7.[Let this point be (xìð,yìð)[The coefficient of í¸2 is[1, which is > 0.[From Exploration 1, we[can conclude that (xìð,yìð)[is a minimum point.[[Apparently xìð is the[axis of symmetry for[the curve of í¸2+6í¸+7,[and it can be  determined[with  sum of roots2.[yìð is found by substitution[of xìð into the  function of[y=í¸2+6í¸+7.[ [We begin the solution by[first locating the roots[ graphically.[- Tap on GraphStrip.[ - Select í¸2+6í¸+7 and[ drag it into the window.[\ GraphStripÌ ÊÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁ aseq_histbNewFolde systemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜ABV#r`` ™O[Once the curve is drawn,[tap on [Analysis]î[G-Solve][î [Root] to see that the[roots are -4.414213562372[and -1.585786437627[[But in finding (xìð,yìð) we[prefer the roots in exact[form. A good technique [to get  the exact roots is[by using quadratic formulae.[Recall our equation is[ í¸2+6í¸+7=0[ Tap to see...\ Use FormulaeË ß R clear_a_zRdoneR1îaR1R6îbR6R7îcR7R-b+b2-4îaîc2îaîRoot1R- 1.585786438R-b-b2-4îaîc2îaîRoot2R- 4.414213562Rsimplify(Root1)R-3+2Rsimplify(Root2)R-3-1î’2RÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁ aseq_histbNewFolde systemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[[The last 2 operations[show the exact roots as[-3+2 and -3-1î’2.[Therefore xìð is, R(-3+2)+(-3-1î’2)2R-3[[ While yìð is xìð2+6xìð+7, orR(-3)2+6(-3)+7R-2[[So the minimum point of[y=í¸2+6í¸+7, or (xìð,yìð),[ is (-3,-2).[[ Exercise 1:[Use the method discussed[to determine the maximum[ or minimum  point of [ y=-2í¸2+5í¸-1.\Use hereÌ ÊÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁ aseq_histbNewFolde systemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜(v@d` ™O\Use hereË ßŒR clear_a_zRdoneR1îaR1R1îbR2R1îcR1RÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁ aseq_histbNewFolde systemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[[Exploration 2:[Lets extend the method[discussed in Example 1[to equation with imaginary[roots. For example the [ equation 5í¸2 -3í¸+9=0.[Our first goal is to find[the max/min point. Lets[call this point (xìñ,yìñ). [[As its coefficient of í¸2[is >0, we conclude that [(xìñ,yìñ) is a minimum point.[[Now tap on CALC strip[and assign appropriate[ values to A,  B and C to[solve 5í¸2 -3í¸+9=0.[\CALCËß´R clear_a_zRdoneR1îAR1R1îBR1R1îCR1R-B+B2-4îAîC2îAîRoot1R-0.5+ 0.8660254038î’îR-B-B2-4îAîC2îAîRoot2R-0.5- 0.8660254038î’îRsimplify(Root1)R-12+12î’îî’3Rsimplify(Root2)R-12î’1+3î’îRÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁ aseq_histbNewFolde systemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[What are our Root1 and[Root2?[Put Root1 and Root2 in[place of ì°, ì± below to[determin e (xìñ,yìñ).[R(ì°)+(ì±)2îxìðR0.5î’P+QR5xìð2-3xìð+9R1.25î’P+Q2-1.5î’P+Q+9[[At this juncture we have[found (xìñ,yìñ). As a final[step we want to verify[graphically that (xìñ,yìñ)[is indeed the minimum of[5í¸2-3í¸+9. [ [- Tap on GraphStrip 2.[- Draw y1=5í¸2-3í¸+9.[- Use [Analysis]î[G-Solve][ î[Min] to locate the[ minimum point.[Does this point consistent[with our (xìñ,yìñ)?[ \ GraphStrip 2ÐÊôGraph2D@ †Graph3DT †LISTSYS`4N†Modify ”$ †STATCALC ¸N†STATSYS À\N†Sequence, †SheetH| †Sheet3DÄ| †SolveEq@†SolveLwrD †SolveUprP †StupFLG1\(†StupList„D †StupPictÈ$ †ViewWindì †y1 H†ä( †ä4 †ä@ †äL †äX †äd †äp †ä| †äˆ †ä” †ä  †ä¬ †ä¸ †äÄ †äÐ †äÜ †ä è †ä!ô †ä" †ä# †ä$ †ä%$ †ä&0 †ä'< †ä(H †ä)T †ä*` †ä+l †ä,x †ä-„ †ä. †ä0œ †ä1¨ †ä2´ †ä3À †ä4Ì †ä5Ø †äEä †äFð †äHü †äI †äJ †äK †äL, †äM8 †äND †äOP †äP\ †äQh †äRt †äS€ †äTŒ †ä]˜ †ä^œ †ä_  †ä`¤ †äa¨ †äb¬ †ä”° †ä•¼ †äÍÈ †äÎÔ †äÐà †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁÎÁ aseq_histbNewFolde systemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁsystemä]listsystemä^]=systemÎÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $  3x5*x^2-3*x+9`P`P``)‡)‡ ™`(1…0qy`#Y‡uYƒ ˜`Q``&1W‰G7 ™P[[ Tap to see...\Exploration NotesáษThe minimum point (xìñ,yìñ) is (0.3,8.55). 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