{"id":2801,"date":"2025-08-13T18:43:00","date_gmt":"2025-08-13T18:43:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2801"},"modified":"2025-10-23T17:48:06","modified_gmt":"2025-10-23T17:48:06","slug":"conics-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/conics-background-youll-need-1\/","title":{"raw":"Conics: Background You'll Need 1","rendered":"Conics: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Complete the square for a quadratic expression<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p data-start=\"190\" data-end=\"488\">Completing the square rewrites a quadratic into a perfect-square form.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><article class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" tabindex=\"-1\" data-turn-id=\"4733c6b5-808b-4f3c-9df5-a1c235a6096c\" data-testid=\"conversation-turn-24\" data-scroll-anchor=\"true\" data-turn=\"assistant\">\r\n<div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)\">\r\n<div class=\"[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\" tabindex=\"-1\">\r\n<div class=\"flex max-w-full flex-col grow\">\r\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-1\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"796ccca8-d7d7-4e95-bb46-437afda78e06\" data-message-model-slug=\"gpt-5-thinking\">\r\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[1px]\">\r\n<div class=\"markdown prose dark:prose-invert w-full break-words light markdown-new-styling\">\r\n<p data-start=\"57\" data-end=\"138\">Given [latex]ax^{2}+bx+c[\/latex] with [latex]a\\neq 0[\/latex], follow these steps to complete the square:<\/p>\r\n\r\n<ol data-start=\"140\" data-end=\"972\">\r\n \t<li data-start=\"140\" data-end=\"272\">\r\n<p data-start=\"143\" data-end=\"272\">Factor out [latex]a[\/latex] from the quadratic and linear terms.<br data-start=\"207\" data-end=\"210\" \/>[latex]\r\nax^{2}+bx+c\r\n=a\\big(x^{2}+\\tfrac{b}{a}x\\big)+c\r\n[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"274\" data-end=\"536\">\r\n<p data-start=\"277\" data-end=\"536\">Add and subtract the square needed to complete the square inside the parentheses.<br data-start=\"358\" data-end=\"361\" \/>The needed square is [latex]\\left(\\tfrac{b}{2a}\\right)^{2}[\/latex].<br data-start=\"428\" data-end=\"431\" \/>[latex]\r\na\\Big(x^{2}+\\tfrac{b}{a}x+\\Big(\\tfrac{b}{2a}\\Big)^{2}-\\Big(\\tfrac{b}{2a}\\Big)^{2}\\Big)+c\r\n[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"538\" data-end=\"715\">\r\n<p data-start=\"541\" data-end=\"715\">Regroup so the first three terms form a perfect square; keep the subtraction inside.<br data-start=\"625\" data-end=\"628\" \/>[latex]\r\na\\Big(\\big(x+\\tfrac{b}{2a}\\big)^{2}-\\Big(\\tfrac{b}{2a}\\Big)^{2}\\Big)+c\r\n[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"717\" data-end=\"972\">\r\n<p data-start=\"720\" data-end=\"972\">Distribute [latex]a[\/latex] and combine constants to finish in vertex form.<br data-start=\"795\" data-end=\"798\" \/>[latex]\r\n\\begin{align}\r\na\\big(x+\\tfrac{b}{2a}\\big)^{2}-a\\Big(\\tfrac{b}{2a}\\Big)^{2}+c\\\\\r\n&amp;=a\\big(x+\\tfrac{b}{2a}\\big)^{2}-\\tfrac{b^{2}}{4a}+c\\\\\r\n&amp;=a(x-h)^{2}+k\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p data-start=\"974\" data-end=\"1054\" data-is-last-node=\"\" data-is-only-node=\"\">Here<br data-start=\"978\" data-end=\"981\" \/>[latex]h=-\\tfrac{b}{2a}[\/latex] and [latex]k=c-\\tfrac{b^{2}}{4a}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/article><\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"490\" data-end=\"561\">Rewrite [latex]2x^{2}-12x+5[\/latex] by completing the square.<\/p>\r\n<p data-start=\"563\" data-end=\"847\">[latex]\r\n\\begin{align}\r\n2x^{2}-12x+5\r\n&amp;=2\\left(x^{2}-6x\\right)+5\\\\\r\n&amp;=2\\left(x^{2}-6x+\\left(\\tfrac{-6}{2}\\right)^{2}-\\left(\\tfrac{-6}{2}\\right)^{2}\\right)+5\\\\\r\n&amp;=2\\left(x^{2}-6x+9-9\\right)+5\\\\\r\n&amp;=2\\left((x-3)^{2}-9\\right)+5\\\\\r\n&amp;=2(x-3)^{2}-18+5\\\\\r\n&amp;=2(x-3)^{2}-13\r\n\\end{align}\r\n[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313909[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313908[\/ohm_question]<\/section>&nbsp;","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Complete the square for a quadratic expression<\/span><\/li>\n<\/ul>\n<\/section>\n<p data-start=\"190\" data-end=\"488\">Completing the square rewrites a quadratic into a perfect-square form.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<article class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" tabindex=\"-1\" data-turn-id=\"4733c6b5-808b-4f3c-9df5-a1c235a6096c\" data-testid=\"conversation-turn-24\" data-scroll-anchor=\"true\" data-turn=\"assistant\">\n<div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)\">\n<div class=\"[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\" tabindex=\"-1\">\n<div class=\"flex max-w-full flex-col grow\">\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-1\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"796ccca8-d7d7-4e95-bb46-437afda78e06\" data-message-model-slug=\"gpt-5-thinking\">\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[1px]\">\n<div class=\"markdown prose dark:prose-invert w-full break-words light markdown-new-styling\">\n<p data-start=\"57\" data-end=\"138\">Given [latex]ax^{2}+bx+c[\/latex] with [latex]a\\neq 0[\/latex], follow these steps to complete the square:<\/p>\n<ol data-start=\"140\" data-end=\"972\">\n<li data-start=\"140\" data-end=\"272\">\n<p data-start=\"143\" data-end=\"272\">Factor out [latex]a[\/latex] from the quadratic and linear terms.<br data-start=\"207\" data-end=\"210\" \/>[latex]ax^{2}+bx+c  =a\\big(x^{2}+\\tfrac{b}{a}x\\big)+c[\/latex]<\/p>\n<\/li>\n<li data-start=\"274\" data-end=\"536\">\n<p data-start=\"277\" data-end=\"536\">Add and subtract the square needed to complete the square inside the parentheses.<br data-start=\"358\" data-end=\"361\" \/>The needed square is [latex]\\left(\\tfrac{b}{2a}\\right)^{2}[\/latex].<br data-start=\"428\" data-end=\"431\" \/>[latex]a\\Big(x^{2}+\\tfrac{b}{a}x+\\Big(\\tfrac{b}{2a}\\Big)^{2}-\\Big(\\tfrac{b}{2a}\\Big)^{2}\\Big)+c[\/latex]<\/p>\n<\/li>\n<li data-start=\"538\" data-end=\"715\">\n<p data-start=\"541\" data-end=\"715\">Regroup so the first three terms form a perfect square; keep the subtraction inside.<br data-start=\"625\" data-end=\"628\" \/>[latex]a\\Big(\\big(x+\\tfrac{b}{2a}\\big)^{2}-\\Big(\\tfrac{b}{2a}\\Big)^{2}\\Big)+c[\/latex]<\/p>\n<\/li>\n<li data-start=\"717\" data-end=\"972\">\n<p data-start=\"720\" data-end=\"972\">Distribute [latex]a[\/latex] and combine constants to finish in vertex form.<br data-start=\"795\" data-end=\"798\" \/>[latex]\\begin{align}  a\\big(x+\\tfrac{b}{2a}\\big)^{2}-a\\Big(\\tfrac{b}{2a}\\Big)^{2}+c\\\\  &=a\\big(x+\\tfrac{b}{2a}\\big)^{2}-\\tfrac{b^{2}}{4a}+c\\\\  &=a(x-h)^{2}+k  \\end{align}[\/latex]<\/p>\n<\/li>\n<\/ol>\n<p data-start=\"974\" data-end=\"1054\" data-is-last-node=\"\" data-is-only-node=\"\">Here<br data-start=\"978\" data-end=\"981\" \/>[latex]h=-\\tfrac{b}{2a}[\/latex] and [latex]k=c-\\tfrac{b^{2}}{4a}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/article>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"490\" data-end=\"561\">Rewrite [latex]2x^{2}-12x+5[\/latex] by completing the square.<\/p>\n<p data-start=\"563\" data-end=\"847\">[latex]\\begin{align}  2x^{2}-12x+5  &=2\\left(x^{2}-6x\\right)+5\\\\  &=2\\left(x^{2}-6x+\\left(\\tfrac{-6}{2}\\right)^{2}-\\left(\\tfrac{-6}{2}\\right)^{2}\\right)+5\\\\  &=2\\left(x^{2}-6x+9-9\\right)+5\\\\  &=2\\left((x-3)^{2}-9\\right)+5\\\\  &=2(x-3)^{2}-18+5\\\\  &=2(x-3)^{2}-13  \\end{align}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313909\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313909&theme=lumen&iframe_resize_id=ohm313909&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313908\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313908&theme=lumen&iframe_resize_id=ohm313908&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2801"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2801\/revisions"}],"predecessor-version":[{"id":4863,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2801\/revisions\/4863"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2801\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2801"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2801"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2801"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2801"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}