We are given the equation:\u22121=V@+n+4-1 = \\sqrt{V@ + n} + 4\u22121=V@+n\u200b+4<\/p>\n\n\n\n
It looks like the original equation contains a typographical error: “\u221a(V@ + n)” is not valid mathematical notation. Let’s assume the correct equation is something more standard, such as:\u22121=x+n+4-1 = \\sqrt{x + n} + 4\u22121=x+n\u200b+4<\/p>\n\n\n\n
Our goal is to solve the equation for xxx<\/strong> and list all valid solutions<\/strong> from least to greatest<\/strong>. If we assume that nnn is a known constant (say, n=5n = 5n=5), then the equation becomes:\u22121=x+5+4-1 = \\sqrt{x + 5} + 4\u22121=x+5\u200b+4<\/p>\n\n\n\n Step 1: Isolate the square root expression<\/strong><\/p>\n\n\n\n Subtract 4 from both sides:\u22121\u22124=x+5\u21d2\u22125=x+5-1 – 4 = \\sqrt{x + 5} \\Rightarrow -5 = \\sqrt{x + 5}\u22121\u22124=x+5\u200b\u21d2\u22125=x+5\u200b<\/p>\n\n\n\n Step 2: Analyze the square root expression<\/strong><\/p>\n\n\n\n Here, we have:x+5=\u22125\\sqrt{x + 5} = -5x+5\u200b=\u22125<\/p>\n\n\n\n This equation has no solution<\/strong>, because the square root of a real number is always non-negative<\/strong> (i.e., a\u22650\\sqrt{a} \\geq 0a\u200b\u22650 for any real number a\u22650a \\geq 0a\u22650). It can never equal a negative number<\/strong> like -5.<\/p>\n\n\n\n There is no real solution<\/strong> to this equation. The square root function cannot produce a negative output, so the equation is not valid for any real value of xxx<\/strong>.<\/p>\n\n\n\n Valid Solutions:<\/strong> To solve a radical equation, our first step is to isolate the radical (the square root in this case). The given equation is:\u22121=x+5+4-1 = \\sqrt{x + 5} + 4\u22121=x+5\u200b+4<\/p>\n\n\n\n We begin by isolating the square root on one side. This is achieved by subtracting 4 from both sides:\u22121\u22124=x+5\u21d2\u22125=x+5-1 – 4 = \\sqrt{x + 5} \\Rightarrow -5 = \\sqrt{x + 5}\u22121\u22124=x+5\u200b\u21d2\u22125=x+5\u200b<\/p>\n\n\n\n Now we examine the resulting expression. The square root function x+5\\sqrt{x + 5}x+5\u200b represents the principal (non-negative) square root. This means it can only yield values greater than or equal to zero. In mathematical terms:x+5\u22650\\sqrt{x + 5} \\geq 0x+5\u200b\u22650<\/p>\n\n\n\n However, in our equation, the square root is set equal to -5. This is not possible, because there is no real number<\/strong> whose square root is negative. Therefore, the equation:\u22125=x+5-5 = \\sqrt{x + 5}\u22125=x+5\u200b<\/p>\n\n\n\n is not valid for any real value of xxx. Attempting to solve such an equation leads to a contradiction of the basic properties of square roots.<\/p>\n\n\n\n Hence, the conclusion is that the equation has no solution<\/strong> in the set of real numbers. In formal math notation, we might write:No Solutionornone\\boxed{\\text{No Solution}} \\quad \\text{or} \\quad \\boxed{\\text{none}}No Solution\u200bornone\u200b<\/p>\n\n\n\n This highlights an important concept: Always check whether the operations in an equation are defined for real numbers<\/strong>. In this case, the equation fails that test due to the square root equaling a negative number<\/p>\n\n\n\n Edulastic Formative and Summative Assessment app.edulastic.com\/student\/assessment\/5f882237e806f400084dag3f\/class\/5f4obeb5sdd6ded177ade83\/uta\/5f8828493130910008e4a9b6\/qid\/13 Question 14\/17 NEXT BOOKMARK Solve the following equation for (List solutions from least to greatest: If an answer box is not needed, write “none” in that box) -1 = \u00e2\u02c6\u0161(V@+n) + 4 Valid Solutions: The Correct Answer and Explanation is: We are given the equation:\u22121=V@+n+4-1 = \\sqrt{V@ + n} […]<\/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-232741","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232741","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=232741"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232741\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232741"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232741"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232741"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step-by-step Solution:<\/h3>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n
\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
none<\/strong><\/p>\n\n\n\n
\n\n\n\nExplanation<\/h3>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-347.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"