{"id":1295,"date":"2012-09-07T22:58:21","date_gmt":"2012-09-07T22:58:21","guid":{"rendered":"http:\/\/testpreparations.com\/help\/?p=1295"},"modified":"2014-12-02T08:32:03","modified_gmt":"2014-12-02T08:32:03","slug":"factoring-two-special-cases","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2012\/09\/07\/factoring-two-special-cases\/","title":{"rendered":"Factoring: Two Special Cases"},"content":{"rendered":"

For a quadratic there are two special cases that factor very quickly and will save time when recognized.<\/p>\n

First case: Difference of Squares<\/h5>\n

A difference of squares occurs when a quadratic has the form<\/p>\n

a2<\/sup>x2<\/sup> \u2013 b2<\/sup> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1)<\/p>\n

where a and b are a positive number. This can then be factored to<\/p>\n

(ax \u2013 b)(ax + b).\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2)<\/p>\n

We can see this is true by expanding (2)<\/p>\n

= (ax \u2013 b)(ax + b)<\/p>\n

= (a2<\/sup>x2<\/sup> + abx \u2013 abx \u2013 b2<\/sup>)<\/p>\n

= (a2<\/sup>x2<\/sup> \u2013 b2<\/sup>)<\/p>\n

And it is shown that a2<\/sup>x2<\/sup> \u2013 b2<\/sup> can be factored into (ax \u2013 b)(ax + b).<\/p>\n

Examples:<\/p>\n

A) x2<\/sup> \u2013 4 = (x \u2013 2)(x + 2)<\/p>\n

We can see that a = 1 and b = 2 in this case.<\/p>\n

B) 81x2<\/sup> \u2013 16 = (9x \u2013 4)(9x + 4) = (3x \u2013 2)(3x + 2)(9x + 4)<\/p>\n

When factoring the first time, another difference of squares is formed and so (9x \u2013 4) is factored again.<\/p>\n

 <\/p>\n

Second Case: Perfect Squares<\/h5>\n

A perfect square occurs when b = 2ac <\/strong>in a quadratic, a2<\/sup> x2 <\/sup>+ bx + c2<\/sup> for some numbers a, b and c. The quadratic can then be factored into<\/p>\n

(ax + c)2<\/sup>.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 (3)<\/p>\n

We can see this is true by expanding (3)<\/p>\n

= (ax + c)2<\/sup><\/p>\n

= a2<\/sup>x2<\/sup> + acx + acx + c2<\/sup><\/p>\n

= a2<\/sup>x2<\/sup> + 2acx + c2<\/sup><\/p>\n

= a2<\/sup>x2<\/sup> + bx + c2<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 because b = 2ac<\/p>\n

 <\/p>\n

And it is shown that if a quadratic, a2<\/sup>x2<\/sup> + bx + c2<\/sup>, satisfies b = 2ac then it can be factored into (ax + c)2<\/sup>.<\/p>\n

Examples:<\/p>\n

A) x2<\/sup> + 6x + 9 = (x + 3)2<\/sup><\/p>\n

We see that a = 1, c = 3 and b = 6 and b = 2ac<\/p>\n

B) 4x2<\/sup> + 12x + 9 = (2x + 3)2<\/sup><\/p>\n

 <\/p>\n

This article was written for you by Jeremie<\/strong>, one of the tutors with Test Prep Academy.<\/span<\/p>\n","protected":false},"excerpt":{"rendered":"

For a quadratic there are two special cases that factor very quickly and will save time when recognized. First case: Difference of Squares A difference of squares occurs when a quadratic has the form a2x2 \u2013 b2 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1) where a and b are a positive number. This can then be factored to […]<\/p>\n","protected":false},"author":6,"featured_media":1754,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2841,3015,3021,2851],"tags":[2725,645,1312,3253],"class_list":["post-1295","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-act","category-math-act","category-mathematics-sat","category-sat","tag-difference-of-squares","tag-factoring","tag-perfect-squares","tag-quadratic"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/1295","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=1295"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/1295\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media\/1754"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=1295"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=1295"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=1295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}