{"id":126857,"date":"2021-09-13T17:12:43","date_gmt":"2021-09-13T11:42:43","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=126857"},"modified":"2021-09-13T17:12:47","modified_gmt":"2021-09-13T11:42:47","slug":"rd-sharma-class-10-solutions-chapter-8-exercise-8-6","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/","title":{"rendered":"RD Sharma Class 10 Solutions Chapter 8 Quadratic Equations Exercise 8.6 (Updated for 2021-22)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-126884\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-8-Exercise-8.6.jpg\" alt=\"RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-8-Exercise-8.6.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-8-Exercise-8.6-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6:&nbsp;<\/strong>At most two roots can be found in a quadratic equation. The type of roots determines whether a quadratic equation can have 1, 2, or even no roots. Exercise 8.6 in <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-for-maths\/\"><strong>RD Sharma Class 10 Solutions<\/strong><\/a> comprises problems that are expressly based on the nature of roots. The <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-quadratic-equations\/\"><strong>RD Sharma Solutions for Class 10 Maths Chapter 8<\/strong><\/a> Exercise 8.6 PDF below contains detailed solutions produced by experts.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69d50598683c5\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" 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href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/#download-rd-sharma-class-10-solutions-chapter-8-exercise-86-free-pdf\" title=\"Download RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF\">Download RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/#access-answers-to-rd-sharma-solutions-class-10-maths-chapter-8-exercise-86-important-question-with-answers\" title=\"Access answers to RD Sharma Solutions Class 10 Maths Chapter 8 Exercise 8.6- Important Question with Answers\">Access answers to RD Sharma Solutions Class 10 Maths Chapter 8 Exercise 8.6- Important Question with Answers<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/#faqs-on-rd-sharma-class-10-solutions-chapter-8-exercise-86\" title=\"FAQs on RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6\">FAQs on RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/#how-is-rd-sharma-class-10-solutions-chapter-8-exercise-86-helpful-for-board-exams\" title=\"How is RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 helpful for board exams?\">How is RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 helpful for board exams?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/#is-the-rd-sharma-class-10-solutions-chapter-8-exercise-86-available-on-the-kopykitab-website\" title=\"Is the RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 available on the Kopykitab website?\">Is the RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 available on the Kopykitab website?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/#where-can-i-get-rd-sharma-class-10-solutions-chapter-8-exercise-86-free-pdf\" title=\"Where can I get RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF?\">Where can I get RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-class-10-solutions-chapter-8-exercise-86-free-pdf\"><\/span>Download RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div id=\"example1\" style=\"text-align: justify;\">&nbsp;<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 800px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solution-Class-10-Chapter-8-Exercise-8.6.pdf\", \"#example1\");<\/script><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solution-Class-10-Chapter-8-Exercise-8.6.pdf\">RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6<\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"access-answers-to-rd-sharma-solutions-class-10-maths-chapter-8-exercise-86-important-question-with-answers\"><\/span>Access answers to RD Sharma Solutions Class 10 Maths Chapter 8 Exercise 8.6- Important Question with Answers<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Question 1.<br>Determine the nature of the roots of following quadratic equations :<br>(i) 2x\u00b2 \u2013 3x + 5 = 0 [NCERT]<br>(ii) 2x\u00b2 \u2013 6x + 3 = 0 [NCERT]<br>(iii)&nbsp;35&nbsp;x\u00b2 \u2013&nbsp;23&nbsp;x + 1 = 0<br>(iv) 3x\u00b2 \u2013 4\u221a3 x + 4 = 0 [NCERT]<br>(v) 3x\u00b2 \u2013 2\u221a6 x + 2 = 0<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1723\/41854101454_b14c2d9d1c_o.png\" alt=\"RD Sharma Class 10 Chapter 8 Quadratic Equations \" width=\"300\" height=\"482\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1735\/41854101314_23833e51a7_o.png\" alt=\"Quadratic Equations Class 10 RD Sharma \" width=\"323\" height=\"472\"><\/p>\n<p>Question 2.<br>Find the values of k for which the roots are real and equal in each of the following equations :<br>(i) kx\u00b2 + 4x + 1 = 0<br>(ii) kx\u00b2 \u2013 2\u221a5 x + 4 = 0<br>(iii) 3x\u00b2 \u2013 5x + 2k = 0<br>(iv) 4x\u00b2+ kx + 9 = 0<br>(v) 2kx\u00b2 \u2013 40x + 25 = 0<br>(vi) 9x\u00b2 \u2013 24x + k = 0<br>(vii) 4x\u00b2 \u2013 3kx +1 = 0<br>(viii) x\u00b2 \u2013 2 (5 + 2k) x + 3 (7 + 10k) = 0<br>(ix) (3k + 1) x\u00b2 + 2(k + 1) x + k = 0<br>(x) kx\u00b2 + kx + 1 = \u2013 4x\u00b2 \u2013 x<br>(xi) (k + 1) x\u00b2 + 2 (k + 3) x + (k + 8) = 0<br>(xii) x\u00b2 \u2013 2kx + 7k \u2013 12 = 0<br>(xiii) (k + 1) x\u00b2 \u2013 2 (3k + 1) x + 8k + 1 = 0<br>(xiv) 5x\u00b2 \u2013 4x + 2 + k (4x\u00b2 \u2013 2x \u2013 1) = 0<br>(xv) (4 \u2013 k) x\u00b2 + (2k + 4) x (8k + 1) = 0<br>(xvi) (2k + 1) x\u00b2 + 2 (k + 3) x (k + 5) = 0<br>(xvii) 4x\u00b2 \u2013 2 (k + 1) x + (k + 4) = 0<br>(xviii) 4x\u00b2 (k + 1) x + (k + 1) = 0<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1749\/42576180511_9f52f98f26_o.png\" alt=\"RD Sharma Class 10 Chapter 8 Quadratic Equations \" width=\"311\" height=\"463\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1732\/42576177891_2b7d608c0f_o.png\" alt=\"Quadratic Equations Class 10 RD Sharma \" width=\"299\" height=\"589\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1745\/41854101744_5cd8d42c5e_o.png\" alt=\"RD Sharma Class 10 Solutions Quadratic Equations \" width=\"288\" height=\"396\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1760\/41854101964_3c0714de53_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 8 Quadratic Equations \" width=\"256\" height=\"450\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1755\/41854102294_c1a285e6a2_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 8 Quadratic Equations \" width=\"362\" height=\"529\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1752\/42576178361_e86ce66f27_o.png\" alt=\"RD Sharma Class 10 Pdf Chapter 8 Quadratic Equations \" width=\"353\" height=\"556\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1758\/42576178751_70c9d88f6a_o.png\" alt=\"RD Sharma Solutions Class 10 Chapter 8 Quadratic Equations \" width=\"357\" height=\"419\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1742\/42576178951_66e4e239e4_o.png\" alt=\"RD Sharma Solutions Class 10 Chapter 8 Quadratic Equations \" width=\"331\" height=\"534\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1741\/42576179231_3d5675d5d4_o.png\" alt=\"Learncbse.In Class 10 Chapter 8 Quadratic Equations \" width=\"362\" height=\"445\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1751\/42576179481_8c7440cd8a_o.png\" alt=\"Learncbse.In Class 10 Chapter 8 Quadratic Equations \" width=\"363\" height=\"526\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1745\/42576179791_c643681b63_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 8 Quadratic Equations \" width=\"364\" height=\"444\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1733\/41854104294_a33ff27d1d_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 8 Quadratic Equations \" width=\"369\" height=\"588\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1751\/42576180061_a7c73b960d_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 8 Quadratic Equations \" width=\"365\" height=\"396\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1757\/42576180181_1be3c0ba8b_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 8 Quadratic Equations \" width=\"354\" height=\"152\"><\/p>\n<p>Question 3.<br>In the following, determine the set of values of k for which the given quadratic equation has real roots :<br>(i) 2x\u00b2 + 3x + k = 0<br>(ii) 2x\u00b2 + x + k = 0<br>(iii) 2x\u00b2 \u2013 5x \u2013 k = 0<br>(iv) kx\u00b2 + 6x + 1 = 0<br>(v) 3x\u00b2 + 2x + k = 0<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1747\/28702416478_8527b41cf9_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 8 Quadratic Equations \" width=\"267\" height=\"458\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1755\/41854104994_b4da6ca766_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 8 Quadratic Equations \" width=\"256\" height=\"224\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1729\/41854105284_5d8d0539cc_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 8 Quadratic Equations \" width=\"256\" height=\"477\"><\/p>\n<p>Question 4.<br>Find the values of k for which the following equations have real and equal roots :<br>(i) x\u00b2- 2(k + 1) x + k\u00b2 = 0 [CBSE 2001C, 2013]<br>(ii) k\u00b2x\u00b2 \u2013 2 (2k \u2013 1) x + 4 = 0 [CBSE 2001C]<br>(iii) (k + 1) x\u00b2 \u2013 2(k \u2013 1) x + 1 = 0 [CBSE 2002C]<br>(iv) x\u00b2 + k(2x + k \u2013 1) + 2 = 0 [CBSE 2017]<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1756\/28702417078_88d331f98f_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 8 Quadratic Equations \" width=\"290\" height=\"283\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1744\/41854105594_d28bec7b54_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 8 Quadratic Equations \" width=\"352\" height=\"371\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1749\/41854105844_d8f3b0bc33_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 8 Quadratic Equations \" width=\"355\" height=\"369\"><\/p>\n<p>Question 5.<br>Find the values of k for which the following equations have real roots<br>(i) 2x\u00b2 + kx + 3 = 0 [NCERT]<br>(ii) kx (x \u2013 2) + 6 = 0 [NCERT]<br>(iii) x\u00b2 \u2013 4kx + k = 0 [CBSE 2012]<br>(iv) kx(x \u2013 2\u221a5 ) + 10 = 0 [CBSE 2013]<br>(v) kx (x \u2013 3) + 9 = 0 [CBSE 2014]<br>(vi) 4x\u00b2 + kx + 3 = 0 [CBSE 2014]<br>Solution:<br>(i) 2x\u00b2 + kx + 3 = 0<br>Here a = 2, b = k, c = 3<br><img src=\"https:\/\/farm2.staticflickr.com\/1757\/28702417708_9f02d29a7d_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 8 Quadratic Equations \" width=\"365\" height=\"551\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1753\/28702417248_88dd52fd74_o.png\" alt=\"Class 10 RD Sharma Chapter 8 Quadratic Equations \" width=\"361\" height=\"425\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1756\/28702417308_c9d12b66ee_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 8 Quadratic Equations \" width=\"219\" height=\"355\"><\/p>\n<p>Question 6.<br>Find the values of k for which the given quadratic equation has real and distinct roots :<br>(i) kx\u00b2 + 2x + 1 = 0<br>(ii) kx\u00b2 + 6x + 1 = 0<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1721\/41854106634_62b7fe6ef4_o.png\" alt=\"Class 10 RD Sharma Chapter 8 Quadratic Equations \" width=\"244\" height=\"411\"><\/p>\n<p>Question 7.<br>For what value of k, (4 \u2013 k) x\u00b2 + (2k + 4) x + (8k + 1) = 0, is a perfect square.<br>Solution:<br>(4 \u2013 k) x\u00b2 + (2k + 4) x + (8k + 1) = 0<br>Here, a = 4 \u2013 k, b = 2k + 4, c = 8k + 1<br><img src=\"https:\/\/farm2.staticflickr.com\/1739\/28702418018_cdb636802f_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 8 Quadratic Equations \" width=\"347\" height=\"344\"><\/p>\n<p>Question 8.<br>Find the least positive value of k for which the equation x\u00b2 + kx + 4 = 0 has real roots.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1754\/41854106924_6f7859f5d9_o.png\" alt=\"RD Sharma 10 Class Solutions Chapter 8 Quadratic Equations \" width=\"265\" height=\"249\"><\/p>\n<p>Question 9.<br>Find the value of k for which the quadratic equation (3k + 1) x\u00b2 + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots.<br>[CBSE 2014]<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1728\/28702418288_e13f508194_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 8 Quadratic Equations \" width=\"342\" height=\"203\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1754\/41854107064_7241696889_o.png\" alt=\"RD Sharma 10 Class Solutions Chapter 8 Quadratic Equations \" width=\"299\" height=\"497\"><\/p>\n<p>Question 10.<br>Find the values of p for which the quadratic equation (2p + 1) x\u00b2 \u2013 (7p + 2) x + (7p \u2013 3) = 0 has equal roots. Also, find these roots.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1736\/41854107524_632981cec6_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 8 Quadratic Equations \" width=\"346\" height=\"363\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1746\/28702418468_475c72585c_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 8 Quadratic Equations \" width=\"316\" height=\"593\"><\/p>\n<p>Question 11.<br>If \u2013 5 is a root of the quadratic equation 2x\u00b2 + px \u2013 15 = 0 and the quadratic equation p(x\u00b2 + x) + k = 0 has equal-roots, find the value of k. [CBSE 2014]<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1730\/28702418868_81645208dd_o.png\" alt=\"RD Sharma Class 10 Solutions Quadratic Equations Ex 8.6 \" width=\"212\" height=\"199\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1729\/28702418808_85b202a722_o.png\" alt=\"RD Sharma Class 10 Solutions Quadratic Equations Ex 8.6 \" width=\"280\" height=\"362\"><\/p>\n<p>Question 12.<br>If 2 is a root of the quadratic equation 3x\u00b2 + px \u2013 8 = 0 and the quadratic equation 4x\u00b2 \u2013 2px + k = 0 has equal roots, find the value of k. [CBSE 2014]<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1745\/28702419108_5d1dd704ef_o.png\" alt=\"Maths RD Sharma Class 10 Solutions Chapter 8 Quadratic Equations \" width=\"267\" height=\"488\"><br>=&gt; 16k = 16<br>k = 16<\/p>\n<p>Question 13.<br>If 1 is a root of the quadratic equation 3x\u00b2 + ax \u2013 2 = 0 and the quadratic equation a(x\u00b2 + 6x) \u2013 b=0 has equal roots, find the value of b.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1747\/28702419258_55195f86dc_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 8 Quadratic Equations \" width=\"296\" height=\"450\"><\/p>\n<p>Question 14.<br>Find the value of p for which the quadratic equation (p + 1) x\u00b2 \u2013 6 (p + 1) x + 3 (p + q) = 0, p \u2260 -1 has equal roots. Hence, find the roots of the equation. [CBSE 2015]<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1743\/28702419908_bf8463f97c_o.png\" alt=\"RD Sharma 10 Chapter 8 Quadratic Equations \" width=\"350\" height=\"221\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1757\/28702419438_5e3fe7205a_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 8 Quadratic Equations \" width=\"294\" height=\"346\"><\/p>\n<p>Question 15.<br>Determine the nature of the roots of following quadratic equations :<br>(i) (x \u2013 2a) (x \u2013 2b) = 4ab<br>(ii) 9a\u00b2b\u00b2x\u00b2 \u2013 24abcdx + 16c\u00b2d\u00b2 = 0, a \u2260 0, b \u2260 0<br>(iii) 2 (a\u00b2 + b\u00b2) x\u00b2 + 2 (a + b) x + 1 = 0<br>(iv) (b + c) x\u00b2 \u2013 (a + b + c) x + a = 0<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1749\/41854109074_bb7f8a2a3c_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 8 Quadratic Equations \" width=\"363\" height=\"466\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1740\/28702420088_e44c170772_o.png\" alt=\"Solution Of RD Sharma Class 10 Chapter 8 Quadratic Equations \" width=\"357\" height=\"414\"><\/p>\n<p>Question 16.<br>Determine the set of values of k for which the given following quadratic equation has real roots :<br>(i) x\u00b2 \u2013 kx + 9 = 0<br>(ii) 2x\u00b2 + kx + 2 = 0<br>(iii) 4x\u00b2 \u2013 3kx +1=0<br>(iv) 2x\u00b2 + kx \u2013 4 = 0<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1746\/41854109304_ef6bafc5a2_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 8 Quadratic Equations \" width=\"243\" height=\"433\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1758\/28702420288_a11b50d202_o.png\" alt=\"Solution Of RD Sharma Class 10 Chapter 8 Quadratic Equations \" width=\"259\" height=\"602\"><\/p>\n<p>Question 17.<br>If the roots of the equation (b \u2013 c) x\u00b2 + (c \u2013 a) x + (a \u2013 b) = 0 are equal, then prove that 2b = a + c. [CBSE 2002C]<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1741\/41854109454_a5dc2c950a_o.png\" alt=\"RD Sharma 10 Solutions Chapter 8 Quadratic Equations \" width=\"341\" height=\"277\"><br>=&gt; a + c = 2b<br>=&gt; 2b = a + c<br>Hence proved.<\/p>\n<p>Question 18.<br>If the roots of the equation (a\u00b2 + b\u00b2) x\u00b2 \u2013 2 (ac + bd) x + (c\u00b2 + d\u00b2) = 0 are equal. prove that&nbsp;ab&nbsp;=&nbsp;cd<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1726\/28702420668_8e1ec13e4b_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 8 Quadratic Equations \" width=\"365\" height=\"408\"><\/p>\n<p>Question 19.<br>If the roots of the equations ax\u00b2 + 2bx + c = 0 and bx\u00b2 \u2013 2\u221aac x + b = 0 are simultaneously real, then prove that b\u00b2 = ac<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1731\/28702420948_a775f09432_o.png\" alt=\"RD Sharma 10 Solutions Chapter 8 Quadratic Equations \" width=\"355\" height=\"298\"><br><img src=\"https:\/\/farm2.staticflickr.com\/1754\/28702420728_02f5eb7211_o.png\" alt=\"RD Sharma Class 10 Maths Chapter 8 Quadratic Equations \" width=\"349\" height=\"103\"><\/p>\n<p>Question 20.<br>If p, q are real and p \u2260 q, then show that the roots of the equation (p \u2013 q) x\u00b2 + 5(p + q) x \u2013 2(p \u2013 q) = 0 are real and unequal.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1754\/41854110094_112310b8a9_o.png\" alt=\"RD Sharma Mathematics Class 10 Pdf Download Free Chapter 8 Quadratic Equations \" width=\"349\" height=\"214\"><\/p>\n<p>Question 21.<br>If the roots of the equation (c\u00b2 \u2013 ab) x\u00b2 \u2013 2 (a\u00b2 \u2013 bc) x + b\u00b2 \u2013 ac = 0 are equal, prove that either a = 0 or a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ c<sup>3<\/sup>&nbsp;= 3abc.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1757\/27706688387_6ff3635f07_o.png\" alt=\"RD Sharma Mathematics Class 10 Pdf Download Free Chapter 8 Quadratic Equations \" width=\"364\" height=\"418\"><\/p>\n<p>Question 22.<br>Show that the equation 2 (a\u00b2 + b\u00b2) x\u00b2 + 2 (a + b) x + 1 = 0 has no real roots, when a \u2260 b.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1736\/27706688477_c230fbb9c4_o.png\" alt=\"RD Sharma Mathematics Class 10 Pdf Download Free Chapter 8 Quadratic Equations \" width=\"346\" height=\"258\"><\/p>\n<p>Question 23.<br>Prove that both the roots of the equation (x \u2013 a) (x \u2013 b) + (x \u2013 b) (x \u2013 c) + (x \u2013 c) (x \u2013 a) = 0 are real but they are equal only when a = b = c.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1731\/28702421738_1b945b4a33_o.png\" alt=\"Class 10 RD Sharma Pdf Chapter 8 Quadratic Equations \" width=\"354\" height=\"563\"><\/p>\n<p>Question 24.<br>If a, b, c are real numbers such that ac \u2260 0, then show that at least one of the equations ax\u00b2 + bx + c = 0 and \u2013 ax\u00b2 + bx + c = 0 has real roots.<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1721\/27706688807_670f4a09f2_o.png\" alt=\"RD Sharma Class 10 Solutions Pdf Free Download Chapter 8 Quadratic Equations \" width=\"348\" height=\"376\"><\/p>\n<p>Question 25.<br>If the equation (1 + m\u00b2) x\u00b2 + 2mcx + (c\u00b2 \u2013 a\u00b2) = 0 has equal roots, prove that c\u00b2 = a\u00b2 (1 + m\u00b2). (C.B.S.E. 1999)<br>Solution:<br><img src=\"https:\/\/farm2.staticflickr.com\/1730\/27706689027_f8e364cb99_o.png\" alt=\"RD Sharma Class 10 Solutions Pdf Free Download Chapter 8 Quadratic Equations \" width=\"356\" height=\"339\"><\/p>\n<p>We have provided complete details of RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6. If you have any queries related to <a href=\"https:\/\/www.cbse.gov.in\/\" target=\"_blank\" rel=\"noopener\"><strong>CBSE<\/strong><\/a>&nbsp;Class 10, feel free to ask us in the comment section below.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-class-10-solutions-chapter-8-exercise-86\"><\/span>FAQs on RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1631525951821\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-is-rd-sharma-class-10-solutions-chapter-8-exercise-86-helpful-for-board-exams\"><\/span>How is RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 helpful for board exams?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>For self-evaluation, RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 provides solutions with thorough descriptions as per term limits specified by the Board. Students will gain valuable experience solving these problems, allowing them to complete the assignment on time.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1631526272525\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"is-the-rd-sharma-class-10-solutions-chapter-8-exercise-86-available-on-the-kopykitab-website\"><\/span>Is the RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 available on the Kopykitab website?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Yes, the PDFs of RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 are available. These solutions are created in a unique method by Kopykitab\u2019s expert faculty.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1631526490531\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"where-can-i-get-rd-sharma-class-10-solutions-chapter-8-exercise-86-free-pdf\"><\/span>Where can I get RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can get RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6 Free PDF from the above article.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Class 10 Solutions Chapter 8 Exercise 8.6:&nbsp;At most two roots can be found in a quadratic equation. The type of roots determines whether a quadratic equation can have 1, 2, or even no roots. Exercise 8.6 in RD Sharma Class 10 Solutions comprises problems that are expressly based on the nature of roots. &#8230; <a title=\"RD Sharma Class 10 Solutions Chapter 8 Quadratic Equations Exercise 8.6 (Updated for 2021-22)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-8-exercise-8-6\/\" aria-label=\"More on RD Sharma Class 10 Solutions Chapter 8 Quadratic Equations Exercise 8.6 (Updated for 2021-22)\">Read more<\/a><\/p>\n","protected":false},"author":238,"featured_media":126884,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3243,9206,73520,4388],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/126857"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/238"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=126857"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/126857\/revisions"}],"predecessor-version":[{"id":127101,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/126857\/revisions\/127101"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/126884"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=126857"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=126857"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=126857"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}