The product of a sum and a difference of the same two terms is equal to a difference of two squares

{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T21:47:12+00:00","modifiedTime":"2016-03-26T21:47:12+00:00","timestamp":"2022-09-14T18:10:29+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"//dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Finding the Sum and Difference of the Same Two Terms","strippedTitle":"finding the sum and difference of the same two terms","slug":"finding-the-sum-and-difference-of-the-same-two-terms","canonicalUrl":"","seo":{"metaDescription":"When distributing binomials over other terms, knowing how to find the sum and difference of the same two terms is a handy shortcut. The sum of any two terms mul","noIndex":0,"noFollow":0},"content":"<p>When distributing binomials over other terms, knowing how to find the sum and difference of the same two terms is a handy shortcut. The sum of any two terms multiplied by the difference of the same two terms is easy to find and even easier to work out — the result is simply the square of the two terms. The middle term just disappears because a term and its opposite are always in the middle.</p>\n<p class=\"Remember\">If you encounter the same two terms and just the sign between them changes, rest assured that the result is the square of those two terms. The second term will always be negative, as in the example, </p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY4Ni5pbWFnZTAucG5n.webp\" width=\"240\" height=\"38\" alt=\"image0.png\"/>\n<p><b><i>Example 1: </i></b>(<i>x</i> – 4)(<i>x</i> + 4)</p>\n<p>You can use the shortcut to do these special distributions. </p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY4Ny5pbWFnZTEucG5n.webp\" width=\"310\" height=\"38\" alt=\"image1.png\"/>\n<p>The second term will always be negative, and a perfect square like the first term: (–4)(+4) = –16.</p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY4OC5pbWFnZTIucG5n.webp\" width=\"272\" height=\"38\" alt=\"image2.png\"/>\n<p><b><i>Example 2</i></b><b><i>: </i></b>(<i>ab</i> – 5)(<i>ab</i> + 5)</p>\n<p>Try the same easy process — multiplying the sum of two terms with their difference — with this slightly more complicated, variable term.</p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY4OS5pbWFnZTMucG5n.webp\" width=\"308\" height=\"38\" alt=\"image3.png\"/>\n<p>The second term is negative, and a perfect square like the first term: 5 = –25.</p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY5MC5pbWFnZTQucG5n.webp\" width=\"314\" height=\"38\" alt=\"image4.png\"/>\n<p><b><i>Example 3</i></b><b><i>: </i></b>[5 + (<i>a</i> – <i>b</i>)][5 – (<i>a</i> – <i>b</i>)]</p>\n<p>This example offers you a chance to work through the sum and difference of various groupings.</p>\n<p>The square of 5 = 25</p>\n<p>The second term is negative, and a perfect square like the first term: </p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY5MS5pbWFnZTUucG5n.webp\" width=\"206\" height=\"38\" alt=\"image5.png\"/>\n<p>Square the binomial and distribute the negative sign, which looks like this: </p>\n<img src=\"//sg.cdnki.com/the-product-of-a-sum-and-a-difference-of-the-same-two-terms-is-equal-to-a-difference-of-two-squares---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzE2NzY5Mi5pbWFnZTYucG5n.webp\" width=\"472\" height=\"112\" alt=\"image6.png\"/>","description":"<p>When distributing binomials over other terms, knowing how to find the sum and difference of the same two terms is a handy shortcut. The sum of any two terms multiplied by the difference of the same two terms is easy to find and even easier to work out — the result is simply the square of the two terms. The middle term just disappears because a term and its opposite are always in the middle.</p>\n<p class=\"Remember\">If you encounter the same two terms and just the sign between them changes, rest assured that the result is the square of those two terms. The second term will always be negative, as in the example, </p>\n<img src=\"//www.dummies.com/wp-content/uploads/167686.image0.png\" width=\"240\" height=\"38\" alt=\"image0.png\"/>\n<p><b><i>Example 1: </i></b>(<i>x</i> – 4)(<i>x</i> + 4)</p>\n<p>You can use the shortcut to do these special distributions. </p>\n<img src=\"//www.dummies.com/wp-content/uploads/167687.image1.png\" width=\"310\" height=\"38\" alt=\"image1.png\"/>\n<p>The second term will always be negative, and a perfect square like the first term: (–4)(+4) = –16.</p>\n<img src=\"//www.dummies.com/wp-content/uploads/167688.image2.png\" width=\"272\" height=\"38\" alt=\"image2.png\"/>\n<p><b><i>Example 2</i></b><b><i>: </i></b>(<i>ab</i> – 5)(<i>ab</i> + 5)</p>\n<p>Try the same easy process — multiplying the sum of two terms with their difference — with this slightly more complicated, variable term.</p>\n<img src=\"//www.dummies.com/wp-content/uploads/167689.image3.png\" width=\"308\" height=\"38\" alt=\"image3.png\"/>\n<p>The second term is negative, and a perfect square like the first term: 5 = –25.</p>\n<img src=\"//www.dummies.com/wp-content/uploads/167690.image4.png\" width=\"314\" height=\"38\" alt=\"image4.png\"/>\n<p><b><i>Example 3</i></b><b><i>: </i></b>[5 + (<i>a</i> – <i>b</i>)][5 – (<i>a</i> – <i>b</i>)]</p>\n<p>This example offers you a chance to work through the sum and difference of various groupings.</p>\n<p>The square of 5 = 25</p>\n<p>The second term is negative, and a perfect square like the first term: </p>\n<img src=\"//www.dummies.com/wp-content/uploads/167691.image5.png\" width=\"206\" height=\"38\" alt=\"image5.png\"/>\n<p>Square the binomial and distribute the negative sign, which looks like this: </p>\n<img src=\"//www.dummies.com/wp-content/uploads/167692.image6.png\" width=\"472\" height=\"112\" alt=\"image6.png\"/>","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"//dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":true,"relatedBook":{"bookId":281938,"slug":"algebra-i-for-dummies-2nd-edition","isbn":"9781119293576","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"//www.amazon.com/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/111929357X-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//catalogimages.wiley.com/images/db/jimages/9781119293576.jpg","width":250,"height":350},"title":"Algebra I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><p><b><b data-author-id=\"8985\">Mary Jane Sterling</b></b> taught mathematics for more than 45 years. She was a professor of mathematics at Bradley University for 35 of those years and continues to teach occasional classes either in person or via distance learning. Sterling is the author of several Dummies algebra and higher-level math titles. She is a graduate of the University of New Hampshire with a master's degree in math education.</p></p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> taught mathematics for more than 45 years. She was a professor of mathematics at Bradley University for 35 of those years and continues to teach occasional classes either in person or via distance learning. Sterling is the author of several Dummies algebra and higher-level math titles. She is a graduate of the University of New Hampshire with a master's degree in math education.</p>","hasArticle":false,"_links":{"self":"//dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"//dummies-api.dummies.com/v2/books/281938"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632219159e389\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632219159ec0f\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":194266},"articleLoadedStatus":"success"},"listState":{"list":{},"objectTitle":"","status":"initial","pageType":null,"objectId":null,"page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{},"filterDataLoadedStatus":"initial","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2022-11-03T10:50:01+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely Script-->\r\n<script src=\"//cdn.optimizely.com/js/10563184655.js\"></script>","enabled":false},{"pages":["all"],"location":"header","script":"<!-- comScore Tag -->\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \"2\", c2: \"15097263\" });(function() {var s = document.createElement(\"script\"), el = document.getElementsByTagName(\"script\")[0]; s.async = true;s.src = (document.location.protocol == \"\" ? \"//sb\" : \"//b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();</script><noscript><img src=\"//sb.scorecardresearch.com/p?c1=2&c2=15097263&cv=2.0&cj=1\" /></noscript>\r\n<!-- / comScore Tag -->","enabled":true},{"pages":["all"],"location":"footer","script":"<!--BEGIN QUALTRICS WEBSITE FEEDBACK SNIPPET-->\r\n<script type='text/javascript'>\r\n(function(){var g=function(e,h,f,g){\r\nthis.get=function(a){for(var a=a+\"=\",c=document.cookie.split(\";\"),b=0,e=c.length;b<e;b++){for(var d=c[b];\" \"==d.charAt(0);)d=d.substring(1,d.length);if(0==d.indexOf(a))return d.substring(a.length,d.length)}return null};\r\nthis.set=function(a,c){var b=\"\",b=new Date;b.setTime(b.getTime()+6048E5);b=\"; expires=\"+b.toGMTString();document.cookie=a+\"=\"+c+b+\"; path=/; \"};\r\nthis.check=function(){var a=this.get(f);if(a)a=a.split(\":\");else if(100!=e)\"v\"==h&&(e=Math.random()>=e/100?0:100),a=[h,e,0],this.set(f,a.join(\":\"));else return!0;var c=a[1];if(100==c)return!0;switch(a[0]){case \"v\":return!1;case \"r\":return c=a[2]%Math.floor(100/c),a[2]++,this.set(f,a.join(\":\")),!c}return!0};\r\nthis.go=function(){if(this.check()){var a=document.createElement(\"script\");a.type=\"text/javascript\";a.src=g;document.body&&document.body.appendChild(a)}};\r\nthis.start=function(){var t=this;\"complete\"!==document.readyState?window.addEventListener?window.addEventListener(\"load\",function(){t.go()},!1):window.attachEvent&&window.attachEvent(\"onload\",function(){t.go()}):t.go()};};\r\ntry{(new g(100,\"r\",\"QSI_S_ZN_5o5yqpvMVjgDOuN\",\"//zn5o5yqpvmvjgdoun-wiley.siteintercept.qualtrics.com/SIE/?Q_ZID=ZN_5o5yqpvMVjgDOuN\")).start()}catch(i){}})();\r\n</script><div id='ZN_5o5yqpvMVjgDOuN'><!--DO NOT REMOVE-CONTENTS PLACED HERE--></div>\r\n<!--END WEBSITE FEEDBACK SNIPPET-->","enabled":false},{"pages":["all"],"location":"header","script":"<!-- Hotjar Tracking Code for //www.dummies.com -->\r\n<script>\r\n (function(h,o,t,j,a,r){\r\n h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)};\r\n h._hjSettings={hjid:257151,hjsv:6};\r\n a=o.getElementsByTagName('head')[0];\r\n r=o.createElement('script');r.async=1;\r\n r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv;\r\n a.appendChild(r);\r\n })(window,document,'//static.hotjar.com/c/hotjar-','.js?sv=');\r\n</script>","enabled":false},{"pages":["article"],"location":"header","script":"<!-- //Connect Container: dummies --> <script src=\"//get.s-onetag.com/bffe21a1-6bb8-4928-9449-7beadb468dae/tag.min.js\" async defer></script>","enabled":true},{"pages":["homepage"],"location":"header","script":"<meta name=\"facebook-domain-verification\" content=\"irk8y0irxf718trg3uwwuexg6xpva0\" />","enabled":true},{"pages":["homepage","article","category","search"],"location":"footer","script":"<!-- Facebook Pixel Code -->\r\n<noscript>\r\n<img height=\"1\" width=\"1\" src=\"//www.facebook.com/tr?id=256338321977984&ev=PageView&noscript=1\"/>\r\n</noscript>\r\n<!-- End Facebook Pixel Code -->","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":287569,"title":"For the Hopeless Romantic","hasSubCategories":false,"url":"/collection/for-the-hopeless-romantic-287569"},{"collectionId":287567,"title":"For the Unabashed Hippie","hasSubCategories":false,"url":"/collection/for-the-unabashed-hippie-287567"},{"collectionId":295430,"title":"Have a Beautiful (and Tasty) Thanksgiving","hasSubCategories":false,"url":"/collection/have-a-wonderful-thanksgiving-295430"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"Article3","path":"/article/academics-the-arts/math/algebra/finding-the-sum-and-difference-of-the-same-two-terms-194266/","hash":"","query":{},"params":{"category1":"academics-the-arts","category2":"math","category3":"algebra","article":"finding-the-sum-and-difference-of-the-same-two-terms-194266"},"fullPath":"/article/academics-the-arts/math/algebra/finding-the-sum-and-difference-of-the-same-two-terms-194266/","meta":{"routeType":"article","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"dropsState":{"submitEmailResponse":false,"status":"initial"},"sfmcState":{"status":"initial"},"profileState":{"auth":{},"userOptions":{},"status":"success"}}