{"id":3743,"date":"2024-01-30T12:29:22","date_gmt":"2024-01-30T06:29:22","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3743"},"modified":"2024-11-07T16:08:16","modified_gmt":"2024-11-07T10:08:16","slug":"quadratic-cubic-equation","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/quadratic-cubic-equation\/","title":{"rendered":"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4, \u09a4\u09cd\u09b0\u09bf\u0998\u09be\u09a4 \u098f\u09ac\u0982 \u099a\u09a4\u09c1\u09b0\u09cd\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Quadratic and Higher Order Equations)"},"content":{"rendered":"

\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u0997\u09a0\u09a8 <\/b>(Formation of quadratic equations):<\/b>\u00a0<\/b><\/span><\/h2>\n

a x^{2}+b x+c=0<\/span> \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0<\/a><\/span> \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09af\u09bc \\alpha \\text { \u098f\u09ac\u0982 } \\beta<\/span><\/span> \u00a0<\/span>\u09b9\u09b2\u09c7 \u098f\u09b0 \u09ae\u09c2\u09b2-\u09b8\u09b9\u0997 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

(i) \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2, <\/span>\u03b1+\u03b2= -\\frac{x^{2} \u098f\u09b0 \u09b8\u09b9\u0997\n\n}{x^{2} \u098f\u09b0 \u09b8\u09b9\u0997\n\n}=-\\frac{b}{a}<\/span><\/span>\u098f\u09ac\u0982 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2, <\/span>\u03b1\u03b2 =\\frac{x \\text { \u09a7\u09cd\u09b0\u09c1\u09ac \u09aa\u09a6 }}{x^{2} \\text { \u098f\u09b0 \u09b8\u09b9\u0997 }}<\/span><\/span><\/p>\n

(ii) \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09b9\u09ac\u09c7, x^{2}<\/span> <\/span>– <\/span>(\u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2) <\/span>x <\/span>+ <\/span>\u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 <\/span>= 0<\/span><\/p>\nx^{2}-(\\alpha+\\beta) x+\\alpha \\beta=0<\/span>\n

 <\/p>\n

\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09be\u09a7\u09be\u09a8<\/b> (The general solution of quadratic equations)<\/b><\/span><\/h2>\n

\u09ac\u09c0\u099c\u0997\u09c1\u09b2\u09bf \u09ae\u09c2\u09b2\u09a6 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7, \u098f\u0995 \u099a\u09b2\u0995\u09c7\u09b0 \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0<\/a><\/span> \u09ac\u09be\u09ae\u09aa\u0995\u09cd\u09b7\u0995\u09c7 \u0989\u09ce\u09aa\u09be\u09a6\u0995\u09c7 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09c7 \u09b8\u09b9\u099c\u09c7\u0987 \u09a4\u09be\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09b8\u09ac \u09b0\u09be\u09b6\u09bf\u09ae\u09be\u09b2\u09be\u0995\u09c7 \u09b8\u09b9\u099c \u0989\u09ce\u09aa\u09be\u09a6\u0995\u09c7 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc \u09a8\u09be\u0964 \u09b8\u09c7 \u099c\u09a8\u09cd\u09af\u00a0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u0995\u09be\u09b0 \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09a8\u09bf\u09ae\u09cd\u09a8\u09b2\u09bf\u0996\u09bf\u09a4 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u099f\u09bf \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u098f\u0995 \u099a\u09b2\u0995 \u09b8\u09ae\u09a8\u09cd\u09ac\u09bf\u09a4 \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0986\u09a6\u09b0\u09cd\u09b6\u09b0\u09c2\u09aa a x^{2}+b x+c=0<\/span><\/span>\u00a0\u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>a,\u00a0b,\u00a0c\u00a0\u2208R<\/span> \u098f\u09ac\u0982<\/span><\/p>\n

\u00a0<\/span>a\u00a0\u2260\u00a00<\/span>.<\/span><\/p>\n

\u2234a^{2} x^{2}+4 a b x+4 a c=0<\/span> [\u0989\u09ad\u09df\u09aa\u0995\u09cd\u09b7\u0995\u09c7<\/span>\u00a04a\u00a0<\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09c7]<\/span><\/p>\n

\u09ac\u09be, (2 a x)^{2}+2 \\cdot 2 a x \\cdot b+b^{2}+4 a c=b^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, (2 a x+b)^{2}=b^{2^{-}} 4 a c<\/span><\/span><\/p>\n

\u09ac\u09be, 2 a x+b=\\pm \\sqrt{b^{2}-4 a c}<\/span><\/span><\/p>\n

\u09ac\u09be, 2 a x=-b \\pm \\sqrt{b^{2}-4 a c}<\/span><\/span><\/p>\n

\u09ac\u09be, x=\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}<\/span><\/span><\/p>\n

\u0985\u09a4\u098f\u09ac, \\frac{-b+\\sqrt{b^{2}-4 a c}}{2 a} \\text { \u098f\u09ac\u0982 } \\frac{-b-\\sqrt{b^{2}-4 a c}}{2 a} \\text { \u09b9\u099a\u09cd\u099b\u09c7 } a x^{2}+b x+c=0<\/span><\/span>\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ae\u09c2\u09b2\u0964\u00a0<\/span><\/p>\n

\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09c2\u09b2-\u09b8\u09b9\u0997<\/b> \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995<\/b> (The basic-coefficient of the quadratic equation)<\/b><\/span><\/h3>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, a x^{2}+b x+c=0<\/span><\/span>\u00a0<\/span>\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df \\alpha \\text {\u098f\u09ac\u0982 } \\beta<\/span><\/span>\u00a0<\/span>. \u09a4\u09be\u09b9\u09b2\u09c7,\u00a0<\/span><\/p>\n\\alpha=\\frac{-b+\\sqrt{b^{2}-4 a c}}{2 a} \\text { \u09b9\u09b2\u09c7 } \\beta=\\frac{-b-\\sqrt{b^{2}-4 a c}}{2 a}<\/span>\n\\alpha+\\beta=\\frac{-b+\\sqrt{b^{2}-4 a c}}{2 a}+\\frac{-b-\\sqrt{b^{2}-4 a c}}{2 a}<\/span>\n

=\\frac{-2 b}{2 a}=\\frac{-b}{a}<\/span> \u098f\u09ac\u0982\u00a0\u00a0\u00a0<\/span><\/p>\n

\\alpha \\beta=\\frac{(-b)^{2}-\\left(\\sqrt{b^{2}-4 a c}\\right)^{2}}{4 a^{2}}=\\frac{-b^{2}-\\left(b^{2}-4 a c\\right)}{4 a^{2}}<\/span> <\/span><\/p>\n

=\\frac{b^{2}-b^{2}+4 a c}{4 a^{2}}=\\frac{4 a c}{4 a^{2}}=\\frac{c}{a}<\/span> <\/span><\/p>\n

\u2234\u00a0<\/span>\u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf\u00a0 \\alpha+\\beta=\\frac{-b}{a}=-\\frac{x \\text { \u098f\u09b0 \u09b8\u09b9\u0997 }}{x^{2} \\text { \u098f\u09b0 \u09b8\u09b9\u0997 }}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 \u03b1 \u03b2 =\\frac{c}{a}<\/span> = -(\u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6)\/(x^{2}\n\n<\/span> \u098f\u09b0 \u09b8\u09b9\u0997<\/span><\/p>\n

\u09af\u09c7\u09ae\u09a8, 4 x^{2}-3 x+2=0<\/span><\/span>\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df \u03b1 \u098f\u09ac\u0982 \u03b2 <\/span>\u00a0\u09b9\u09b2\u09c7, \\alpha+\\beta=-\\frac{-3}{4}=\\frac{3}{4}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 \\alpha \\beta=\\frac{2}{4}=\\frac{1}{2}<\/span><\/span><\/p>\n

1. \u09af\u09c7 \u09b6\u09b0\u09cd\u09a4 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 a_{1} x^{2}+b_{1} x+c_{1}=0<\/span><\/b>\u00a0<\/span>\u098f\u09ac\u0982 a_{2} x^{2}+b_{2} x+c_{2}=0<\/span><\/b>\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u098f\u0995\u099f\u09bf \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ae\u09c2\u09b2 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ae\u09c2\u09b2\u099f\u09bf <\/span>; \u09a4\u09be\u09b9\u09b2\u09c7 <\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0989\u09ad\u09df \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0987 \u09b8\u09bf\u09a6\u09cd\u09a7 \u09b9\u09ac\u09c7,<\/span><\/p>\na_{1} \\alpha^{2}+b_{1} \\alpha+c_{1}=0<\/span>\n

\u098f\u09ac\u0982 a_{2} \\alpha^{2}+b_{2} \\alpha+c_{2}=0<\/span><\/span><\/p>\n

(i) \u0993 (ii) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09ac\u099c\u09cd\u09b0\u0997\u09c1\u09a3\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n\\frac{\\mathbf{a}^{2}}{\\mathrm{~b}_{1} \\mathrm{c}_{2}-\\mathrm{b}_{2} \\mathrm{c}_{1}}=\\frac{\\mathbf{a}}{\\mathrm{c}_{1} \\mathrm{a}_{2}-\\mathrm{c}_{2} \\mathrm{a}_{1}}=\\frac{1}{\\mathrm{a}_{1} \\mathrm{~b}_{2}-\\mathrm{a}_{2} \\mathrm{~b}_{1}}<\/span>\n

\\therefore \\alpha^{2}=\\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}<\/span>\u00a0 \u00a0 [\u09e7\u09ae \u0993 \u09e9\u09df \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u09b9\u09a4\u09c7]\u00a0<\/span><\/p>\n

\\alpha=\\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}<\/span>\u00a0 \u00a0[\u09e8\u09df \u0993 \u09e9\u09df \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u09b9\u09a4\u09c7]<\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\left(\\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\right)=\\left(\\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\right)^{2}<\/span><\/span><\/p>\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce,\\left(b_{1} c_{2}-b_{2} c_{1}\\right)\\left(a_{1} b_{2}-a_{2} b_{1}\\right)=\\left(c_{1} a_{2}-c_{2} a_{1}\\right)^{2}<\/span><\/span>\u00a0\u00a0<\/span>\u09af\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09b6\u09b0\u09cd\u09a4\u0964<\/span><\/p>\n

\u09ac\u09bf.\u09a6\u09cd\u09b0. <\/b>\u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ae\u09c2\u09b2 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09af\u09bc \u09a8\u09bf\u09ae\u09cd\u09a8\u09b2\u09bf\u0996\u09bf\u09a4 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09b9\u099c\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc,<\/span><\/p>\n\\left(b_{1} c_{2}-b_{2} c_{1}\\right)\\left(a_{1} b_{2}-a_{2} b_{1}\\right)=\\left(c_{1} a_{2}-c_{2} a_{1}\\right)^{2}<\/span>\n

[<\/span>\u09aa\u09cd\u09b0\u09a5\u09ae<\/span>\u00a0<\/span>\u09b0\u09be\u09b6\u09bf\u09b0<\/span>\u00a0<\/span>\u09b9\u09b0<\/span>\u00a0\u00d7\u00a0<\/span>\u09b6\u09c7\u09b7<\/span>\u00a0<\/span>\u09b0\u09be\u09b6\u09bf\u09b0<\/span>\u00a0<\/span>\u09b9\u09b0<\/span>\u00a0<\/span>=<\/span> (<\/span>\u09ae\u09a7\u09cd\u09af\u09b0\u09be\u09b6\u09bf\u09b0<\/span>\u00a0<\/span>\u09b9\u09b0)<\/span>2<\/span>]<\/span><\/p>\n

2. \u09af\u09c7 \u09b6\u09b0\u09cd\u09a4 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 a_{1} x^{2}+b_{1} x+c_{1}=0<\/span><\/b>\u00a0<\/span>\u098f\u09ac\u0982 a_{2} x^{2}+b_{2} x+c_{2}=0<\/span><\/b>\u00a0<\/span>\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0989\u09ad\u09df \u09ae\u09c2\u09b2\u0987 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ae\u09c2\u09b2\u099f\u09bf <\/span> \u0993 <\/span>; \u09a4\u09be\u09b9\u09b2\u09c7 \u09ae\u09c2\u09b2 \u0993 \u09b8\u09b9\u0997\u09c7\u09b0 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7,\u00a0<\/span><\/p>\n

\u09aa\u09cd\u09b0\u09a5\u09ae \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \\left(a_{1} x^{2}+b_{1} x+c_{1}=0\\right)<\/span><\/span>\u00a0<\/span>\u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span>\u00a0<\/span> \\left.\\begin{array}{l}\n\n\\alpha+\\beta=\\frac{-b_{1}}{a_{1}} \\\\\n\n\\text { <b>\u098f\u09ac\u0982<\/b> } \\alpha \\beta=\\frac{c_{1}}{a_{1}}\n\n\\end{array}\\right\\}<\/span><\/span>\u00a0 \u00a0<\/span>\u2026..(i)<\/span><\/p>\n

\u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09df \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \\left(a_{2} x^{2}+b_{2} x+c_{2}=0\\right)<\/span><\/span>\u00a0<\/span>\u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span> \\left.\\begin{array}{l}\n\n\\alpha+\\beta=\\frac{-b_{2}}{a_{2}} \\\\\n\n\\text { <b>\u098f\u09ac\u0982<\/b> } \\alpha \\beta=\\frac{c_{2}}{a_{2}}\n\n\\end{array}\\right\\}<\/span><\/span>\u00a0 \u2026..(ii)<\/span><\/p>\n

(i) \u0993 (ii) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

\\frac{-b_{1}}{a_{1}}=\\frac{-b_{2}}{a_{2}} \\text { <b>\u098f\u09ac\u0982<\/b> } \\frac{c_{1}}{a_{1}}=\\frac{c_{2}}{a_{2}}<\/span><\/span><\/p>\n

\u09ac\u09be, \\frac{a_{1}}{a_{2}}=\\frac{b_{1}}{b_{2}} \\text { <b>\u098f\u09ac\u0982<\/b> } \\frac{a_{1}}{a_{2}}=\\frac{c_{1}}{c_{2}}<\/span><\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, <\/span>\\frac{a_{1}}{a_{2}}=\\frac{b_{1}}{b_{2}}=\\frac{c_{1}}{c_{2}}<\/span><\/span>\u00a0\u09af\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09b6\u09b0\u09cd\u09a4\u0964<\/span><\/p>\n

\u09aa\u09c3\u09a5\u09be\u09df\u0995<\/b> (Discriminant):<\/b>\u00a0<\/b><\/span><\/h2>\n

a x^{2}+b x+c=0<\/span>; <\/span>(a,\u00a0b,\u00a0c\u00a0\u2208\u00a0R\u00a0<\/span>\u098f\u09ac\u0982 <\/span>a<\/span> 0<\/span>) \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df\u00a0 =\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}<\/span>\u00a0<\/span> \u098f\u09b0 \u09a7\u09b0\u09a3 \u0993 \u09aa\u09cd\u09b0\u0995\u09c3\u09a4\u09bf \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2\u09c7\u09b0 \u09ad\u09bf\u09a4\u09b0\u09c7\u09b0 \u0985\u0982\u09b6 b^{2}-4 a c<\/span><\/span> \u00a0<\/span>\u098f\u09b0 \u0989\u09aa\u09b0 \u09a8\u09bf\u09b0\u09cd\u09ad\u09b0\u09b6\u09c0\u09b2\u0964 \u09a4\u09be\u0987, b^{2}-4 a c<\/span> <\/span>\u0995\u09c7 \u0989\u0995\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09aa\u09c7\u09b0 \u09aa\u09c3\u09a5\u09be\u09df\u0995 (discriminant) \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09aa\u09c3\u09a5\u09be\u09df\u0995\u09c7\u09b0<\/b> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09ad\u09c7\u09a6\u09c7 \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09a7\u09b0\u09a8 \u0993 \u09aa\u09cd\u09b0\u0995\u09c3\u09a4\u09bf<\/b> (The type and nature of the two basic equations of quadratic equations in the case of discriminant):<\/b><\/p>\n

1<\/strong>. b^{2}-4 a c>0<\/span><\/span>\u00a0<\/strong>\u09b9\u09b2\u09c7 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u0993 \u0985\u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7\u0964<\/b> \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 y=f(x)=a x^{2}+b x+c<\/span><\/span>\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be\u099f\u09bf (\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4) <\/span>x<\/span>– \u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964\"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4<\/span><\/p>\n

\"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4<\/p>\n

2. b^{2}-4 a c=0<\/span><\/b>\u00a0\u09b9\u09b2\u09c7 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df \u09ac\u09be\u09b8\u09cd\u09a4\u09ac, \u09ae\u09c1\u09b2\u09a6 \u0993 \u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7\u0964<\/b> \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 x=\\frac{-b}{2 a}, \\frac{-b}{2 a}<\/span> \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 y=f(x)=a x^{2}+b x+c<\/span> \u00a0\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be\u099f\u09bf x-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4<\/p>\n

\"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4<\/p>\n

3. b^{2}-4 a c<0<\/span> <\/strong>\u09b9\u09b2\u09c7 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df \u099c\u099f\u09bf\u09b2 \u0993 \u0985\u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7\u0964<\/b> \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09af\u09bc \u09b8\u09ac\u09b8\u09ae\u09af\u09bc \u09a6\u09c1\u0987\u099f\u09bf \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u099c\u099f\u09bf\u09b2 \u09ac\u09be \u0995\u09be\u09b2\u09cd\u09aa\u09a8\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09df\u0964 \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, y=f(x)=a x^{2}+b x+c<\/span><\/span>\u00a0\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be\u099f\u09bf x-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u099b\u09c7\u09a6 \u09ac\u09be \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09c7 \u09a8\u09be\u0964<\/span><\/p>\n

\"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4<\/p>\n

\"\u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4<\/p>\n

4<\/strong>. a, b, c \u2208 Q <\/strong>\u098f\u09ac\u0982 b^{2}-4 a c<\/span><\/strong>\u00a0\u09aa\u09c2\u09b0\u09cd\u09a3\u09ac\u09b0\u09cd\u0997 \u09b9\u09b2\u09c7 \u09ae\u09c2\u09b2\u09a6\u09cd\u09ac\u09df \u09ae\u09c1\u09b2\u09a6 \u0993 \u0985\u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7\u0964<\/b><\/p>\n

\u00a0\u00a0\u00a0\u00a0 \\therefore b^{2}-4 a c=0<\/span><\/span><\/p>\n

 <\/p>\n

\u09a4\u09cd\u09b0\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09c2\u09b2-\u09b8\u09b9\u0997 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 <\/b>(Relation between roots and coefficients of cubic equations):<\/b><\/span><\/h2>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, a x^{3}+b x^{2}+c x+d=0<\/span><\/span>\u00a0\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09c2\u09b2\u09a4\u09cd\u09b0\u09df <\/span>\u03b1,\u00a0\u00a0\u03b2,\u00a0\u00a0\u03b3<\/span>\u00a0<\/span><\/p>\n

\u09a4\u09be\u09b9\u09b2\u09c7, x^{3}+\\frac{b}{a} x^{2}+\\frac{c}{a} x+\\frac{d}{a} \\equiv(x-\\alpha)(x-\\beta)(x-\\gamma)<\/span>\u00a0<\/span><\/p>\n

\u09ac\u09be, x^{3}+\\frac{b}{a} x^{2}+\\frac{c}{a} x+\\frac{d}{a} \\equiv x^{3}-(\\alpha+\\beta+\\gamma) x^{2}+(\\alpha \\beta+\\beta \\gamma+\\gamma \\alpha) x-\\alpha \\beta \\gamma<\/span><\/span><\/p>\n

\u2234<\/span> \\therefore \\alpha+\\beta+\\gamma=-\\frac{b}{a}=(-1)^{1} \\frac{b}{a}<\/span>\u00a0<\/span><\/p>\n\\alpha \\beta+\\beta \\gamma+\\gamma \\alpha=\\frac{c}{a}=(-1)^{2} \\cdot \\frac{c}{a^{\\prime}}<\/span>\n

\u098f\u09ac\u0982 \\alpha \\beta \\gamma=-\\frac{d}{a}(-1)^{3} \\frac{d}{a} \\quad\\left[x^{3}, x^{2}, x\\right.<\/span><\/span>\u00a0<\/span>\u0993 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u09aa\u09a6\u0997\u09c1\u09b2\u09bf \u09b8\u09ae\u09c0\u0995\u09c3\u09a4 \u0995\u09b0\u09c7]<\/span><\/p>\n

\u09a4\u09cd\u09b0\u09bf\u0998\u09be\u09a4<\/b> \u098f\u09ac\u0982 \u099a\u09a4\u09c1\u09b0\u09cd\u0998\u09be\u09a4 \u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/b> \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09be\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u099c\u09a8\u09c0\u09df \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf<\/b> (Necessary method to solve equations consisting of cubic and quadrilaterals):<\/b><\/span><\/h3>\n