[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-143080-105":59,"doc-detail-143080-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","math-problems-quadratic-function-equations-and-geometry-area-solutions","Math Problems - Quadratic Function Equations and Geometry Area Solutions","","A set of math practice questions focuses on solving for variables that satisfy the same solution set after a function transformation, using substitution into a quadratic definition and comparing resulting equations. Subsequent problems apply factoring of parabolas to find intercepts, derive rectangle dimensions from vertex coordinates, and compute triangle area from circle-line intersections via the distance formula for coordinates. The final challenge uses constrained averages and ratio limits to maximize a second-highest net worth given upper bounds.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/math-problems-quadratic-function-equations-and-geometry-area-solutions/143080/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/math-problems-quadratic-function-equations-and-geometry-area-solutions/143080.png","ImageObject",300,407,{"name":92,"@type":93},"Patrick","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you determine which equation has the same solution set after the transformation f(x+2)=4?","Question",{"text":113,"@type":114},"Substitute x+2 into f(x)=x^2+x+2, set the result equal to 4, and simplify to an equation of the form x^2+5x+k=0. Match the constant term k to the answer choice.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the area of rectangle ABCD found when a line segment is tangent at the parabola vertex?",{"text":118,"@type":114},"Factor the parabola to find x-intercepts A and B and use their midpoint to get the vertex x-coordinate. Substitute the x-coordinate into the factored form to get the vertex y-value (height), then multiply width (difference of intercepts) by height.",{"name":120,"@type":111,"acceptedAnswer":121},"How do you compute the area of triangle ABC from the circle x^2+y^2=10 and line y=2x?",{"text":122,"@type":114},"Use the intersection condition by substituting y=2x into the circle equation to solve for x at point C, then compute y. With A at (0,0) and B on the x-axis, use half base times height to get the triangle area.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},143080,1787692677,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145},549758146520,"https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470","If f(x)=x²+x+2,then which of theequations below has the same solution setas f(x+2)=4?  \n(A)x²+5x+2=0  \n(B)x²+5x+4=0  \n(C)x²+5x+6=0  \n(D)x²+3x+2=0  \n(E)x²+3x+6=0  \nIf f(x)=x²+x+2,then which of theequations below has the same solution setas f(x+2)=4?  \n(A)x²+5x+2=0  \n(B)x²+5x+4=0  \n(C)x²+5x+6=0  \n(D)x²+3x+2=0  \n(E)x²+3x+6=0  \nThis problem is(purposely)stated in a strangeway.They are really asking:“If you plug(x+2)intothe function definition and subtract 4,what to youget?\"Here’s the math:  \nf(x+2)=(x+2)²+(x+2)+2=x²+5x+8=4.  \nSo now we havex²+5x+8=4 or x²+5x+4=0and that's the answer(B).  \nX  \nNote:Figure not drawn to scale.  \n   \nThe parabolay=4x²-12x+5 crossesthe x-axis at points A and B as shownabove.Line segment CD in rectangleABCD is tangent to the parabola at itsvertex.What is the area of rectangleABCD?(A)8(B)10(C)12(D)15(E)20  \nSolution  \nX  \nNote:Figure not drawn to scale.  \n   \nThe parabolay=4x²-12x+5 crossesthe x-axis at points A and B as shownabove.Line segment CD in rectangleABCD is tangent to the parabola at itsvertex.What is the area of rectangleABCD?  \ny=4x²-12x+5=(2x-1)(2x-5).  \nSo when xis 1/2,y is zero.Also whenxis5/2,y iszero.Point Ais(1/2,0)and point B is(5/2,0).  \nYou've got the length of AB now-it's 2 of course;you just subtract the x values.But you need theheight of the rectangle.  \nThe x-coordinate of the vertex is right in themiddle of points Aand B,so that's 3/2.If you plugit into the equation fory(it's easiest to use thefactored form)you get the y-coordinate of thevertex to be-4.  \nThe height is 4 and the width is 2,so the area is 8and the answer is(A).  \nX  \n-4  \n-4  \nThe circle defined by the equationx²+y²=10 is shown in the diagramabove.The line y=2x intersects thecircle at point C and forms the  \nhypotenuse of right triangle ABC.If pointAis at the origin and point B is on theX-axis,what is the area of triangle ABC?(A)2(B)2√2(C)3√2(D)√10/2(E)none of these  \nSolution  \nX  \nThe circle defined by the equationx²+y²=10 is shown in the diagramabove.The line y=2x intersects thecircle at point C and forms the  \nhypotenuse of right triangle ABC.If pointA is at the origin and point B is on thex-axis,what is the area of triangle ABC?  \nSince point Cis an intersection point,bothequations are true for this point.So you can plug y=2x into the circle equation and get,  \nx²+(2x)²=10→5x²=10→x=√2→y=2√2.  \nWith both the x and y coordinates for point C,youhave the base and height of your triangle and canplug in one-half base times height for the area:  \nThe answer is 2 or(A).  \nThe curve f(x)=x²-2 is shown above.A line(not shown)with positive slope mintersects the curve at two points whosecoordinates are(a,b)and (3,c).If a\u003C0,which of the following contains threepossible values of the slope m?  \nI.  1.5,2.5,3.5  \nI.  1,2,3ⅢI.2,4,6(A)none(B)I,only(C)II,only(D)Ⅲ,only(E)I,,and Ⅲ  \nI.  1.5,2.5,3.5  \nⅡ.1,2,3Ⅲ.2,4,6  \nX  \nThe curve f(x)=x²-2 is shown above.A line (not shown)with positive slope mintersects the curve at two points whosecoordinates are(a,b)and (3,c).If a\u003C0,which of the following contains threepossible values of the slope m?  \nThe point(3,c)is of course just(3,7)if you plug 3into the function.The other point(a,b)issomewhere on the left side of the curve because ais negative.The bottom of the curve is(0,-2).  \nThe line from(0,-2)to(3,7)has a slope of 3.Theslope of the line from(a,b)to(3,7)must be LESSthan 3.Any slope between zero and 3(notincluding zero or 3)is possible.  \nNone of the choices has three possible slopes,sothe answer is(A).  \nTen club members have an average networth of 1million dollars each and a totalnet worth of 10 million dollars.No one inthe club has more than twice as muchmoney as any other member.The richestmember has 1.8million dollars.What isthe largest possible net worth for thesecond-richest member of the club?  \n(A)1.4 million dollars  \n(B)1.3 million dollars  \n(C)1.2 million dollars  \n(D)1.1 million dollars  \n(E)1.0 million dollars  \nTen club members have an average networth of 1million dol","cbCaieBByiAX3FRB","https://ap.wps.com/l/cbCaieBByiAX3FRB","pdf",50481946,151,"English","# Quadratic function transformation - same solution set\n# Parabola factoring - rectangle area\n# Circle-line intersection - triangle area\n# Slope constraints from curve secant lines\n# Inequalities and ratios - maximizing net worth","[{\"question\":\"How do you determine which equation has the same solution set after the transformation f(x+2)=4?\",\"answer\":\"Substitute x+2 into f(x)=x^2+x+2, set the result equal to 4, and simplify to an equation of the form x^2+5x+k=0. Match the constant term k to the answer choice.\"},{\"question\":\"How is the area of rectangle ABCD found when a line segment is tangent at the parabola vertex?\",\"answer\":\"Factor the parabola to find x-intercepts A and B and use their midpoint to get the vertex x-coordinate. Substitute the x-coordinate into the factored form to get the vertex y-value (height), then multiply width (difference of intercepts) by height.\"},{\"question\":\"How do you compute the area of triangle ABC from the circle x^2+y^2=10 and line y=2x?\",\"answer\":\"Use the intersection condition by substituting y=2x into the circle equation to solve for x at point C, then compute y. With A at (0,0) and B on the x-axis, use half base times height to get the triangle area.\"}]","Math Problems - Quadratic Function Equations and Geometry Area Solutions | PDF",381]