{"id":11877,"date":"2015-08-21T06:06:03","date_gmt":"2015-08-21T06:06:03","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=11877"},"modified":"2018-02-14T16:45:18","modified_gmt":"2018-02-14T16:45:18","slug":"radical-expressions-the-golden-ratio","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/radical-expressions-the-golden-ratio\/","title":{"rendered":"Radical expressions:  The golden ratio"},"content":{"rendered":"<h1>The tutor introduces the golden ratio.<\/h1>\n<p>The golden ratio is defined as x\/y, where<\/p>\n<p>x\/y = (x+y)\/x, x>y, y>0<\/p>\n<p>Muliplying both sides by xy gives<\/p>\n<p>x^2=xy+y^2<\/p>\n<p>which leads to<\/p>\n<p>x^2-xy-y^2=0<\/p>\n<p>Using the quadratic formula to solve for x, we imagine<\/p>\n<p>x=(-b &#177; (b^2-4ac)^0.5)\/(2a)<\/p>\n<p>based on<\/p>\n<p>ax^2+bx+c=0<\/p>\n<p>In our particular equation is<\/p>\n<p>a=1, b=-y, c=-y^2<\/p>\n<p>which leads to<\/p>\n<p>x=(-(-y)&#177;((-y)^2-4(1)(-y^2))^0.5)\/(2*1)<\/p>\n<p>Simplifying, we get to<\/p>\n<p>x=(y+(5y^2)^0.5)\/2 or x=(y-(5y^2)^0.5)\/2<\/p>\n<p>which becomes<\/p>\n<p>x=(y+y(5)^0.5)\/2 or (y-y(5)^0.5)\/2<\/p>\n<p>Since x>0,<\/p>\n<p>x=(y+y(5)^0.5)\/2 only.<\/p>\n<p>Continuing to simplify, we get<\/p>\n<p>x=y(1+5^0.5)\/2<\/p>\n<p>So the golden ratio, x\/y, is<\/p>\n<p>x\/y = (y(1+5^0.5)\/2)\/y = (1+5^0.5)\/2 &#8776; 1.618<\/p>\n<p>I&#8217;ll be discussing the golden ratio further in future posts:)<\/p>\n<p>Source:<\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Golden_ratio\">Wikipedia<\/a><\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The tutor introduces the golden ratio. The golden ratio is defined as x\/y, where x\/y = (x+y)\/x, x>y, y>0 Muliplying both sides by xy gives x^2=xy+y^2 which leads to x^2-xy-y^2=0 Using the quadratic formula to solve for x, we imagine &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/radical-expressions-the-golden-ratio\/\"> <span class=\"screen-reader-text\">Radical expressions:  The golden ratio<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[1005,1006],"class_list":["post-11877","post","type-post","status-publish","format-standard","hentry","category-math","tag-golden-ratio","tag-solving-for-the-golden-ratio"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/11877","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=11877"}],"version-history":[{"count":28,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/11877\/revisions"}],"predecessor-version":[{"id":29722,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/11877\/revisions\/29722"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=11877"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=11877"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=11877"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}