#### Cumulative Assessment

Exercises 1 The information whether the discriminant is positive, negative, or zero can be used to determine the number of real solutions of a quadratic equation. However, it works both ways — the number of real solutions tells us whether the discriminant is positive, negative, or zero.DiscriminantNumber of Solutions PositiveTwo real solutions NegativeNo real solutions ZeroOne real solutionTo determine whether the discriminants of the given equations are positive, negative, or zero, we will determine the number of solutions of each equation. f(x)=0g(x)=0h(x)=0j(x)=0​ The solutions of these equations, if any, are the x-intercepts of the related functions, f, g, h, and j, respectively. Let's take a look at the given graph.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["Solution21713_4335870_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try { var b=mlg.board([-7.5,6.5,7.5,-6.5],{desktopSize:"medium",style:"usa"}); b.xaxis(2,1); b.yaxis(2,1); var el = {};el.func1 = b.func("1.4*x^2+6.1*x+7.7"); el.func2 = b.func("-2*x^2-3*x+2",{strokeColor:"red"}); el.func3 = b.func("x^2-4*x+4",{strokeColor:"green"}); el.func4 = b.func("-0.32*x^2",{strokeColor:"darkviolet"}); el.legend1 = b.legend(el.func1,[-3.91,5.03],"f"); el.legend2 = b.legend(el.func2,[-2.63,-3.93],"g"); el.legend3 = b.legend(el.func3,[3.54,2.07],"h"); el.legend4 = b.legend(el.func4,[3.14,-2.89],"j");mlg.af("showA",function() { b.remove(el); el.func1 = b.func("1.4*x^2+6.1*x+7.7"); el.func2 = b.func("-2*x^2-3*x+2",{strokeColor:"red"}); el.func3 = b.func("x^2-4*x+4",{strokeColor:"green"}); el.func4 = b.func("-0.32*x^2",{strokeColor:"darkviolet"}); el.legend1 = b.legend(el.func1,[-3.91,5.03],"f"); el.legend2 = b.legend(el.func2,[-2.63,-3.93],"g"); el.legend3 = b.legend(el.func3,[3.54,2.07],"h"); el.legend4 = b.legend(el.func4,[3.14,-2.89],"j"); });mlg.af("showB",function() { b.remove(el); el.func1 = b.func("1.4*x^2+6.1*x+7.7"); el.legend1 = b.legend(el.func1,[-3.91,5.03],"f"); });mlg.af("showC",function() { b.remove(el); el.func2 = b.func("-2*x^2-3*x+2",{strokeColor:"red"}); el.legend2 = b.legend(el.func2,[-2.63,-3.93],"g"); el.p1 = b.point(-2,0); el.p2 = b.point(0.5,0); el.flag1 = b.flag(el.p1,"x\\text{-intercept}",152,2.3); el.flag2 = b.flag(el.p2,"x\\text{-intercept}",27,2.3); });mlg.af("showD",function() { b.remove(el); el.func3 = b.func("x^2-4*x+4",{strokeColor:"green"}); el.func4 = b.func("-0.32*x^2",{strokeColor:"darkviolet"}); el.legend3 = b.legend(el.func3,[3.54,2.07],"h"); el.legend4 = b.legend(el.func4,[3.14,-2.89],"j"); el.p1 = b.point(2,0,{fillColor:"green"}); el.p2 = b.point(0,0,{fillColor:"darkviolet"}); el.flag1 = b.flag(el.p1,"x\\text{-intercept}",19,3.9); el.flag2 = b.flag(el.p2,"x\\text{-intercept}",161,3.4); });} catch( e ) { mw.hook("ml.error.report").fire(e, { tags: { module: "jsxg", graph: "Solution21713_4335870_l" } }); console.error(e); } if(b) {b.board.update()} }; mw.loader.using('ext.MLJSXGraph', function() { window.ml.jsxgraph.buildRealGraph("Solution21713_4335870_l", "Solution21713_4335870_p", 1, code); }); } ); } window.JXQtable["Solution21713_4335870_l"] = true;Allwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1192738778_904360145').on('touchstart mousedown', function () { try { mlg.cf("showA"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );fwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1207289197_1730546574').on('touchstart mousedown', function () { try { mlg.cf("showB"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );gwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn1113131442_1865803475').on('touchstart mousedown', function () { try { mlg.cf("showC"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );h and jwindow.RLQ = window.RLQ || []; window.RLQ.push( function () { (function ($) {$('#jsxbtn644397787_566050357').on('touchstart mousedown', function () { try { mlg.cf("showD"); } catch(e) { mw.log.error(e); } }); })(jQuery); } );We can see that f has no x-intercepts, g has two x-intercepts, and h and j have one x-intercept each. Therefore, f(x)=0 has no real solutions, g(x)=0 has two real solutions, and h(x)=0 and j(x)=0 have one real solution each. Let's make a table to see what it means in terms of the dicriminants!EquationNumber of SolutionsDiscriminant f(x)=0No real solutionsNegative g(x)=0Two real solutionsPositive h(x)=0One real solutionZero j(x)=0One real solutionZero