What is the partial fraction decomposition of 7x^2-6x+9\/3x(4x^2+9)<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The correct answer is: 3×1\u200b+4×2+9x\u22122\u200b<\/strong><\/mark><\/p>\n\n\n\n To find the partial fraction decomposition of the expression (\\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)}), we start by identifying the factors in the denominator, which are (3x) and (4x^2 + 9). The first factor, (3x), is a linear term, and (4x^2 + 9) is a quadratic term that cannot be factored over the reals.<\/p>\n\n\n\n The general form for the partial fraction decomposition will look like this:<\/p>\n\n\n\n [ Here, (A), (B), and (C) are constants that we need to determine.<\/p>\n\n\n\n To combine the right side into a single fraction, we need a common denominator:<\/p>\n\n\n\n [ Expanding the numerator gives:<\/p>\n\n\n\n [ Now we equate this to the numerator of the original expression:<\/p>\n\n\n\n [ From this, we can create a system of equations by comparing coefficients:<\/p>\n\n\n\n From equation 3, we find:<\/p>\n\n\n\n [ Substituting (A = 1) into equation 1:<\/p>\n\n\n\n [ From equation 2:<\/p>\n\n\n\n [ Now substituting back the values of (A), (B), and (C):<\/p>\n\n\n\n [ Thus, the partial fraction decomposition of (\\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)}) is:<\/p>\n\n\n\n [ This decomposition allows us to simplify the original expression, making it easier to integrate or analyze in further calculations.<\/p>\n","protected":false},"excerpt":{"rendered":" What is the partial fraction decomposition of 7x^2-6x+9\/3x(4x^2+9) The Correct Answer and Explanation is : The correct answer is: 3×1\u200b+4×2+9x\u22122\u200b To find the partial fraction decomposition of the expression (\\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)}), we start by identifying the factors in the denominator, which are (3x) and (4x^2 + 9). The first factor, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-149700","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149700","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=149700"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149700\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=149700"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=149700"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=149700"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Setting Up the Partial Fractions<\/h3>\n\n\n\n
\\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)} = \\frac{A}{3x} + \\frac{Bx + C}{4x^2 + 9}
]<\/p>\n\n\n\nStep 2: Combine the Right Side<\/h3>\n\n\n\n
\\frac{A}{3x} + \\frac{Bx + C}{4x^2 + 9} = \\frac{A(4x^2 + 9) + (Bx + C)(3x)}{3x(4x^2 + 9)}
]<\/p>\n\n\n\nStep 3: Expand and Equate<\/h3>\n\n\n\n
A(4x^2 + 9) + (Bx + C)(3x) = 4Ax^2 + 9A + 3Bx^2 + 3Cx = (4A + 3B)x^2 + 3Cx + 9A
]<\/p>\n\n\n\n
(4A + 3B)x^2 + 3Cx + 9A = 7x^2 – 6x + 9
]<\/p>\n\n\n\nStep 4: Set Up the System of Equations<\/h3>\n\n\n\n
\n
Step 5: Solve for the Constants<\/h3>\n\n\n\n
A = 1
]<\/p>\n\n\n\n
4(1) + 3B = 7 \\implies 3B = 3 \\implies B = 1
]<\/p>\n\n\n\n
3C = -6 \\implies C = -2
]<\/p>\n\n\n\nStep 6: Write the Partial Fraction Decomposition<\/h3>\n\n\n\n
\\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)} = \\frac{1}{3x} + \\frac{x – 2}{4x^2 + 9}
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n
\\frac{1}{3x} + \\frac{x – 2}{4x^2 + 9}
]<\/p>\n\n\n\n