{"id":4503,"date":"2022-01-24T10:04:19","date_gmt":"2022-01-24T02:04:48","guid":{"rendered":"https:\/\/www.study-together.com\/edu\/dse-maths-core\/fundamental_algebra\/1-%e4%b8%80%e5%85%83%e4%ba%8c%e6%ac%a1%e6%96%b9%e7%a8%8b-%e5%88%a4%e5%88%a5%e5%bc%8f\/"},"modified":"2022-04-06T04:00:38","modified_gmt":"2022-04-05T20:00:38","slug":"discriminant","status":"publish","type":"page","link":"https:\/\/www.study-together.com\/edu\/dse-maths-core\/fundamental_algebra\/quadratic_equation\/discriminant\/","title":{"rendered":"\u5224\u5225\u5f0f"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"4503\" class=\"elementor elementor-4503\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4f989a04 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4f989a04\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-66 elementor-top-column elementor-element elementor-element-54cb3494\" data-id=\"54cb3494\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-30d99766 elementor-widget elementor-widget-text-editor\" data-id=\"30d99766\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h2>1.5 \u7406\u89e3\u4e8c\u6b21\u65b9\u7a0b\u7684 \u5224\u5225\u5f0f \u8207\u5176\u6839\u7684\u6027\u8cea\u4e4b\u95dc\u4fc2 \uff08Understand the Relations between the Discriminant of a Quadratic Equation and the Nature of its Roots\uff09<\/h2><h3>1.5.1 \u5224\u5225\u5f0f \u8207\u5176\u6839\u7684\u6027\u8cea\u4e4b\u95dc\u4fc2<\/h3><p>\u8981100%\u660e\u767d \u5224\u5225\u5f0f \u8207\u5176\u4e8c\u6b21\u65b9\u7a0b\u7684\u6839\u7684\u6027\u8cea\uff0c\u5927\u5bb6\u8981\u5148\u660e\u767d\u958b\u65b9\u6839 \\(\\sqrt{\\quad}\\)\u5605\u6027\u8cea\u3002<\/p><ul><li>\u9996\u5148\u8acb\u5927\u5bb6\u7528\u8a08\u6578\u6a5f\u8a08\u5413\u4ee5\u4e0b\u5605\u6578\u503c\uff1a<ul><li>\\(2+\\sqrt{9}=\\underline{\\quad\\quad }\\).<br \/>\\(2-\\sqrt{9}=\\underline{\\quad\\quad }.\\)<\/li><li>\\(2+\\sqrt{0}=\\underline{\\quad\\quad }.\\)<br \/>\\(2-\\sqrt{0}=\\underline{\\quad\\quad }.\\)<\/li><li>\\(2+\\sqrt{-3}=\\underline{\\quad\\quad }.\\)<br \/>\\(2-\\sqrt{-3}=\\underline{\\quad\\quad }.\\)<br \/>\u4e0a\u9762\u5169\u984c\u56e0\u70ba\u6709\u8a08\u5230\u201cMath Error\u201d\u4fc2\u6b63\u5e38\u5605\uff01<\/li><li>\u6211\u8ad7\u5927\u5bb6\u90fd\u6ce8\u610f\u5230\u958b\u65b9\u6839\u5165\u9762\u5605\u6578\u503c\u6703\u9ede\u5f71\u97ff\u8a08\u51fa\u569f\u5605\u7b54\u6848\u3002<\/li><\/ul><\/li><\/ul><p>\u6839\u64da\u6c42\u6839\u516c\u5f0f\uff0c\\(\\quad\\quad x = \\tfrac{-b\\pm\\sqrt{b^2-4ac}}{2a}\\)<\/p><ul><li>\u5047\u5982\u6211\u54cb\u8a2d\u00a0 \\(\\Delta=b^2-4ac\\)\uff0c\u5481\u65b9\u7a0b\u5605<span style=\"color: #ff0000;\">\u5169\u500b\u6839<\/span>\u5c31\u4fc2\uff1a<br \/>\\(\\quad x=\\dfrac{-b+\\sqrt{\\Delta}}{2a} \\quad \u6216 \\quad x=\\dfrac{-b-\\sqrt{\\Delta}}{2a}\\)<\/li><li>\u7559\u610f\uff0c\u5982\u679c\uff1a<ul><li>\u2206\uff1d0\uff0c\u201c\\(-b+\\sqrt{\\Delta}\\)\u201d\u540c\u201c\\(-b-\\sqrt{\\Delta}\\)\u201d\u90fd\u6703\u8b8a\u6210\u201c\uff0db\u201d\u3002\u5481\u5605\u8a71<ul><li>\u5169\u500b\u6839\u90fd\u6703\u7b49\u65bc\u201c\uff0db \/ 2a\u201d\u3002<\/li><\/ul><\/li><li>\u2206\uff1e0\uff0c\u201c\\(-b+\\sqrt{\\Delta}\\)\u201d\u540c\u201c\\(-b-\\sqrt{\\Delta}\\)\u201d\u5c31\u6703\u5514\u4e00\u6a23<ul><li>\u5373\u4fc2\u5169\u500b\u6839\u6703\u5514\u76f8\u7b49\u3002<\/li><\/ul><\/li><li>\u2206\uff1c0\uff0c\\sqrt{\\Delta} \u4fc2\u8a08\u5514\u5230\u5605<ul><li>\u6211\u54cb\u6703\u8a71\u201c\u65b9\u7a0b\u6c92\u6709\u5be6\u6839\u201d\uff08\u5be6\u6839=\u5be6\u6578\u7684\u6839\uff0c\u4f8b\u59821.1, \\(sqrt{2}\\)\u3002<\/li><\/ul><\/li><\/ul><\/li><\/ul><h4>\u7e3d\u7d50\uff1a<\/h4><ul><li>\u201c\u6839\u7684\u6027\u8cea\u201d\u4fc2\u8b1b\u7dca\u5169\u500b\u6839\u662f\u5426\u5b58\u5728\uff0c\u540c\u57cb\u4fc2\u5514\u4fc2\u4e00\u6a23<\/li><li>\\(\\Delta=b^2-4ac\\) \u4fc2\u53eb\u505a\u201c\u5224\u5225\u5f0f\u201d\uff08discriminant\uff09<\/li><li>\u6839\u64da\u5224\u5225\u5f0f\u5605\u5927\u7d30(\u6b63\u3001\u8ca0\u3001\u96f6)\uff0c\u6211\u54cb\u53ef\u4ee5\u77e5\u9053\u6839\u5605\u6027\u8cea\uff1a<ul><li><span style=\"color: #ff0000;\">\u0394 &gt; 0\u6642\uff0c\u65b9\u7a0b\u6709\u5169\u500b\u4e0d\u540c\u7684\u5be6\u6839 (2 unequal real roots)<\/span><\/li><li><span style=\"color: #ff0000;\">\u0394 = 0\u6642\uff0c\u65b9\u7a0b\u6709\u5169\u500b\u76f8\u540c\u7684\u5be6\u6839 (2 equal real roots)<\/span><\/li><li><span style=\"color: #ff0000;\">\u0394 &lt; 0\u6642\uff0c\u65b9\u7a0b\u6c92\u6709\u5be6\u6839 (no real roots)<\/span><\/li><\/ul><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6490f98 elementor-widget elementor-widget-text-editor\" data-id=\"6490f98\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h3>1.5.2 \u5e38\u898b\u8003\u6cd5<\/h3><p>\u55baDSE\u8a66\u984c\u5165\u9762\uff0c\u7d93\u5e38\u6703\u51fa\u73fe\u201c\u5df2\u77e5 x<sup>2<\/sup> + 5x + k = 0 \u6c92\u6709\u5be6\u6839\uff0c\u6c42k\u7684\u7bc4\u570d\u3002\u201d\u6216\u76f8\u985e\u4f3c\u5605\u554f\u984c\u3002<\/p><p>\u5176\u5be6\u8981\u505a\u4e00D\u90fd\u5514\u96e3!\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-12548cc elementor-widget elementor-widget-accordion\" data-id=\"12548cc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1921\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1921\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">\u4f8b\uff1a \u5df2\u77e5 \\(x^2+5x+k=0\\) \u6c92\u6709\u5be6\u6839\uff0c\u6c42k\u7684\u7bc4\u570d\u3002<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1921\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1921\"><ul><li>\u601d\u8003\u904e\u7a0b:<ul><li>\u898b\u5230\u201c\u6c92\u6709\u5be6\u6839\u201d\uff0c\u5927\u5bb6\u5c31\u8981\u8ad7\u8d77\u201c\u0394 &lt; 0 \u201d<\/li><li>\u5373\u3000\\(5^2\u20134(1)(k)&lt;0\\)<\/li><\/ul><\/li><li>\u6240\u4ee5\u500b\u7b54\u6848\u53ef\u4ee5\u5481\u5beb\uff1a<br \/>\\(\\begin{align}<br \/>\\quad\\quad \u56e0\u70ba\u6c92\u6709\u5be6\u6839\uff0c\\quad\\Delta&amp;&lt;0 \\\\<br \/>5^2 \u2013 4(1)(k)&amp;&lt;0\\\\<br \/>25&amp;&lt;4k\\\\<br \/>k&amp;&gt;6.25<br \/>\\end{align}\\)<\/li><\/ul><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-6723d148 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6723d148\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-344e0f3b\" data-id=\"344e0f3b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1a0313e elementor-widget elementor-widget-text-editor\" data-id=\"1a0313e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><i class=\"far fa-arrow-alt-circle-left\" aria-hidden=\"true\"><\/i>\u00a0<a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/fundamental_algebra\/quadratic_equation\/solve_by_formula\/\">\u4e8c\u6b21\u65b9\u7a0b\u516c\u5f0f<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-6fe0e492\" data-id=\"6fe0e492\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-1173629a\" data-id=\"1173629a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2c8d02e6 elementor-widget elementor-widget-text-editor\" data-id=\"2c8d02e6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: right;\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/fundamental_algebra\/quadratic_equation\/problems_related_to_quadratic_equation\/\">\u4e8c\u6b21\u65b9\u7a0b\u7684\u61c9\u7528\u984c<\/a>\u00a0 <i class=\"far fa-arrow-alt-circle-right\" aria-hidden=\"true\"><\/i><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1bd821d8\" data-id=\"1bd821d8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-499346e elementor-widget elementor-widget-sidebar\" data-id=\"499346e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"sidebar.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"sidebar-box widget-oceanwp-custom-menu custom-menu-widget clr\"><style type=\"text\/css\">.ocean_custom_menu-5 > ul > li > a, .custom-menu-widget .ocean_custom_menu-5 .dropdown-menu .sub-menu li a.menu-link{font-size:16px;line-height:17px;}.custom-menu-widget .ocean_custom_menu-5.oceanwp-custom-menu > ul.click-menu .open-this{font-size:16px;}<\/style><h4 class=\"widget-title\">DSE\u6578\u5b78 &#8211; \u5fc5\u4fee\u90e8\u5206\u3010\u76ee\u9304\u3011<\/h4><div class=\"oceanwp-custom-menu clr ocean_custom_menu-5 left dropdown-hover\"><ul id=\"menu-dsemaths_core\" class=\"dropdown-menu sf-menu\"><li  id=\"menu-item-1744\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-1744\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/course-analysis\/\" class=\"menu-link\">\u8ab2\u7a0b\u5206\u6790 (Core)<\/a><\/li>\n<li  id=\"menu-item-1745\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children dropdown menu-item-1745\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/\" class=\"menu-link\">\u5514\u597d\u4ee5\u70ba\u5514\u99db\u8003 &#8211; \u76ee\u9304 <span class=\"nav-arrow fa fa-angle-down\"><\/span><\/a>\n<ul class=\"sub-menu\">\n\t<li  id=\"menu-item-4067\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4067\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/solving-equation\/\" class=\"menu-link\">1. \u89e3\u65b9\u7a0b<\/a><\/li>\n\t<li  id=\"menu-item-4068\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4068\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/polynomials\/\" class=\"menu-link\">2. \u591a\u9805\u5f0f<\/a><\/li>\n\t<li  id=\"menu-item-4069\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4069\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/laws-of-indices\/\" class=\"menu-link\">3. \u6307\u6578\u5b9a\u5f8b<\/a><\/li>\n\t<li  id=\"menu-item-4070\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4070\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/factorization\/\" class=\"menu-link\">4. \u56e0\u5f0f\u5206\u89e3<\/a><\/li>\n\t<li  id=\"menu-item-4071\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4071\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/formulae\/\" class=\"menu-link\">5. \u516c\u5f0f<\/a><\/li>\n\t<li  id=\"menu-item-4072\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4072\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/percentage\/\" class=\"menu-link\">6. \u767e\u5206\u6cd5<\/a><\/li>\n\t<li  id=\"menu-item-4073\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4073\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/rate-and-ratio\/\" class=\"menu-link\">7. \u7387\u53ca\u6bd4<\/a><\/li>\n\t<li  id=\"menu-item-4074\" class=\"menu-item 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id=\"menu-item-4079\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4079\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/geometry\/\" class=\"menu-link\">12. \u6f14\u7e79\u63a8\u7406\u5e7e\u4f55<\/a><\/li>\n\t<li  id=\"menu-item-4080\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4080\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/trigonometry\/\" class=\"menu-link\">13. \u4e09\u89d2\u6bd4<\/a><\/li>\n\t<li  id=\"menu-item-4081\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4081\"><a href=\"https:\/\/www.study-together.com\/edu\/dse-maths-core\/junior_summary\/shapes\/\" class=\"menu-link\">14. \u5e73\u9762\u53ca\u7acb\u9ad4\u5716\u5f62<\/a><\/li>\n\t<li  id=\"menu-item-4082\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-4082\"><a 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