(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 84001, 2038] NotebookOptionsPosition[ 73013, 1873] NotebookOutlinePosition[ 73965, 1901] CellTagsIndexPosition[ 73886, 1896] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"4820b046-9bd3-42f2-931d-7338fda02c54"], Cell["Using the Discriminant to solve Quadratic Problems", "Title", CellChangeTimes->{{3.732487804154851*^9, 3.732487821220138*^9}}, TextAlignment->Center, FontWeight->"Bold",ExpressionUUID->"2ad2632a-3134-4ede-ba2d-c6dad600f362"], Cell["\<\ This notebook covers multiple ways in which the discriminant can be used in \ the topic of Quadratics. 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the graph will have two \ x-intercepts. \ \>", "Text", CellFrame->{{0.5, 3}, {3, 0.5}}, CellChangeTimes->{{3.732922415557773*^9, 3.732922449722116*^9}}, Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"27a411e8-e6fe-431c-a916-2aa16a9371f8"], Cell[TextData[{ "\nFind the x-intercepts to ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", RowBox[{"(", "x", ")"}]}], "=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"10", "x"}], "+", "7"}]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "c92b581b-2c2e-4b90-9e10-ab0be73e3722"], "." }], "Text", CellChangeTimes->{ 3.732488039482971*^9, {3.73248811408436*^9, 3.732488115485263*^9}, 3.732488870655184*^9},ExpressionUUID->"6adf5822-cbc1-470c-b66e-\ 4b3e95971e2b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"0", "\[Equal]", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"10", "x"}], "+", "7"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.7324879417443247`*^9, 3.7324879421486473`*^9}, { 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", "Text", CellFrame->{{0.5, 3}, {3, 0.5}}, CellChangeTimes->{{3.732922578523939*^9, 3.732922592902672*^9}}, Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"c59bed88-2bbe-4406-b97c-652497973ded"] }, Open ]], Cell[CellGroupData[{ Cell["", "Subtitle",ExpressionUUID->"7e3e7d1d-bbe4-48af-878a-cc0afbe382e5"], Cell[TextData[{ "Question 3: Consider the quadratic ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", RowBox[{"(", "x", ")"}]}], "=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"b", " ", "x"}], "+", "7"}]}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "9e7118ad-60d3-4606-9e46-41f79d6ea3f0"] }], "Subtitle", CellChangeTimes->{{3.732489050512761*^9, 3.73248906993541*^9}},ExpressionUUID->"ca0a9e8c-9467-4412-921a-\ 0eca4580343e"], Cell["\<\ Find the discriminant.\ \>", "Text", CellChangeTimes->{{3.732488950577833*^9, 3.732488965735783*^9}, { 3.732489087474489*^9, 3.7324890960262938`*^9}},ExpressionUUID->"88c4675f-d49a-4fdd-a4db-\ 44962e9f8bcb"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Discriminant", "[", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", "bx", " ", "+", "7"}], ",", " ", "x"}], "]"}]], "Input", CellChangeTimes->{{3.7324890986605062`*^9, 3.73248909909333*^9}, { 3.7329222709599*^9, 3.732922305021842*^9}}, CellLabel->"In[28]:=",ExpressionUUID->"3f2fb4cd-a3aa-4f32-b829-cda4c3c2b166"], Cell[BoxData[ RowBox[{ RowBox[{"-", "4"}], " ", RowBox[{"(", RowBox[{"7", "+", "bx"}], ")"}]}]], "Output", CellChangeTimes->{3.732922306464744*^9, 3.732922383360078*^9, 3.7329224258835363`*^9}, CellLabel->"Out[28]=",ExpressionUUID->"9583240a-3857-433d-b90b-752259ec501f"] }, Open ]], Cell[TextData[{ "\nHence, use your discriminant equation to find the value(s) of b when \ there is only one x-intercept. 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You may need more or less input cells. Up to you!\n\n1. Find \ an expression when equation 1 and equation 2 are equal. \n2. Put all of the \ terms on one side. Your equation should be in the form of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["ax", "2"], "+", "bx", "+", "c"}], "=", "0."}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "b1774fb7-bc5f-4f5d-9438-b110e6a54d38"], "\n3. Find the discriminant of your equation from step 2. You do not need \ the = 0 there. \n4. Your answer should be " }], "Text", CellChangeTimes->{{3.732490544843148*^9, 3.73249060013623*^9}, { 3.732490662954898*^9, 3.7324906635628223`*^9}, {3.73249072468777*^9, 3.73249096899816*^9}},ExpressionUUID->"8a0c239c-7cd4-42ec-a023-\ 0e2f9ffca81b"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.732493586285727*^9, 3.73249358672138*^9}},ExpressionUUID->"bc079807-7436-4ebd-baca-\ 0fd4db2a1ff7"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.7324909376565027`*^9, 3.732490938425097*^9}},ExpressionUUID->"0d71dc00-eab9-486d-a9d7-\ 2ce289700456"], Cell["\<\ Hence, use your discriminant to find the value of p when f(x) and g(x) have \ two points of intersection. If you get stuck, go back to slide 1.\ \>", "Text", CellChangeTimes->{{3.732490976116994*^9, 3.732491017344718*^9}, { 3.732491206883115*^9, 3.732491216376957*^9}},ExpressionUUID->"6b5a0a77-71ad-4f8b-89f7-\ 0740980aa850"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.7324910209932337`*^9, 3.73249102135937*^9}},ExpressionUUID->"c95919e0-d129-490e-a69a-\ 2b0bc568321a"], Cell["\<\ Answer (don\[CloseCurlyQuote]t forget to write it in interval notation):\ \>", "Text", CellChangeTimes->{{3.732487971780427*^9, 3.732487977398251*^9}, { 3.732488152917238*^9, 3.732488168487804*^9}, {3.732488879159886*^9, 3.732488891826021*^9}, {3.732489334645575*^9, 3.732489365839841*^9}, { 3.732489462617144*^9, 3.732489489128594*^9}, {3.732489541443523*^9, 3.7324896169611483`*^9}, {3.732490372326356*^9, 3.7324903798380013`*^9}, { 3.732490576386732*^9, 3.732490590410408*^9}, {3.7324910367972097`*^9, 3.732491056141616*^9}},ExpressionUUID->"0e13faca-8a0f-47e5-87c2-\ e6d37e34d258"], Cell["", "Text", CellFrame->{{0.5, 3}, {3, 0.5}}, Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"c847f862-71bd-4a49-9f14-bccf2ed6fe57"], Cell["\<\ Use your discriminant to find the value of p when g(x) is tangent to f(x).\ \>", "Text", CellChangeTimes->{{3.732491062541184*^9, 3.732491086498343*^9}},ExpressionUUID->"7bbc796b-fc6b-4872-960f-\ d954c04acff0"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.732491088691555*^9, 3.732491089014968*^9}},ExpressionUUID->"8d791d64-2e09-43cf-b3d3-\ 79365745999b"], Cell["\<\ Answer (don\[CloseCurlyQuote]t forget to write it in interval notation):\ \>", "Text", CellChangeTimes->{{3.732487971780427*^9, 3.732487977398251*^9}, { 3.732488152917238*^9, 3.732488168487804*^9}, {3.732488879159886*^9, 3.732488891826021*^9}, {3.732489334645575*^9, 3.732489365839841*^9}, { 3.732489462617144*^9, 3.732489489128594*^9}, {3.732489541443523*^9, 3.7324896169611483`*^9}, {3.732490372326356*^9, 3.7324903798380013`*^9}, { 3.732490576386732*^9, 3.732490590410408*^9}, {3.7324910367972097`*^9, 3.732491056141616*^9}},ExpressionUUID->"bfe227a3-ac62-4f83-a76f-\ 218f1f86e3fb"], Cell["", "Text", CellFrame->{{0.5, 3}, {3, 0.5}}, Background->RGBColor[ 1, 0.85, 0.85],ExpressionUUID->"d52a7e82-851b-43f0-a713-97e90ea85268"], Cell["\<\ Plot f(x) and g(x) (with your value of p), to check that g(x) is tangent to \ f(x). Don\[CloseCurlyQuote]t forget to use AspectRatio in your command line \ or you will not see that it is tangent.\ \>", "Text", CellChangeTimes->{{3.732491109262361*^9, 3.732491164717361*^9}},ExpressionUUID->"e5213b40-3412-4953-944d-\ c01103bf9c1f"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.732491168595194*^9, 3.7324911688721952`*^9}},ExpressionUUID->"029a9872-4d45-478a-aecb-\ d7aaf8eb6328"], Cell["\<\ Use your discriminant to find the value(s) of p when g(x) and f(x) never \ intersect.\ \>", "Text", CellChangeTimes->{{3.732491062541184*^9, 3.732491086498343*^9}, { 3.732491254389989*^9, 3.73249127141282*^9}},ExpressionUUID->"48f04ff6-aa62-47ef-8083-\ 9016d4954b63"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.732491088691555*^9, 3.732491089014968*^9}},ExpressionUUID->"e9d637a7-2d5f-4622-93f1-\ 9bdf292b50c9"], Cell["\<\ Answer (don\[CloseCurlyQuote]t forget to write it in interval notation):\ \>", "Text", CellChangeTimes->{{3.732487971780427*^9, 3.732487977398251*^9}, { 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Show that the equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "+", RowBox[{"2", RowBox[{"(", RowBox[{"p", "+", "1"}], ")"}], "x"}], "+", "p"}], TraditionalForm]], ExpressionUUID->"38a4121a-e776-4414-8753-eb33cf064b67"], " = 0 always has real roots." }], "Text", CellChangeTimes->{{3.732494233755912*^9, 3.7324942608209677`*^9}},ExpressionUUID->"ab800240-3103-4ce5-b023-\ ad4a037ac96f"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.732494263894562*^9, 3.732494264526956*^9}},ExpressionUUID->"f345b524-9e21-49d0-ae91-\ 367588dc0b4a"], Cell[CellGroupData[{ Cell[TextData[{ "\nQuestion 7: Find the relationship between p and q if the roots of the \ equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["px", "2"], "+", RowBox[{"q", " ", "x"}], "+", "1"}], "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "69abfa0d-e7b4-4c83-b0ec-4561085f0e6b"], " are equal." }], "Subtitle", CellChangeTimes->{{3.732493786662704*^9, 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