{"id":497,"date":"2024-01-11T22:59:27","date_gmt":"2024-01-11T14:59:27","guid":{"rendered":"https:\/\/big-tan.top\/?p=497"},"modified":"2024-02-13T12:05:17","modified_gmt":"2024-02-13T04:05:17","slug":"%e4%bb%a3%e6%95%b0%e5%bc%8f%e6%b1%82%e5%80%bc%e9%97%ae%e9%a2%98%e6%8a%80%e5%b7%a7%e9%9b%86%e9%94%a6%ef%bc%88%e4%ba%94%ef%bc%89","status":"publish","type":"post","link":"https:\/\/itantan.eu.org\/?p=497","title":{"rendered":"\u4ee3\u6570\u5f0f\u6c42\u503c\u95ee\u9898\u6280\u5de7\u96c6\u9526\uff08\u4e94\uff09"},"content":{"rendered":"\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-97606a7 stk-block-background\" data-block-id=\"97606a7\"><style>.stk-97606a7{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-97606a7:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-97606a7-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ac5e8d7\" data-v=\"4\" data-block-id=\"ac5e8d7\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-ac5e8d7-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-ac5e8d7-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-97b7cd9\" data-block-id=\"97b7cd9\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e00\uff1a\u82e5$x^2+y^2=7\uff0cx^3+y^3=10$\uff0c\u6c42$x+y$\u7684\u503c\u3002\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u96be\u5ea6\u592a\u5927\uff0c\u9700\u8981\u6280\u5de7\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7531\u4e8e\u5b58\u57283\u6b21\u9879\uff0c\u8054\u7acb\u5c55\u5f00\u6c42\u89e3\u6bd4\u8f83\u590d\u6742\uff0c\u9700\u8981\u5c06$x+y$\u770b\u6210\u6574\u4f53\uff0c\u8bbe\u6cd5\u6784\u9020\u5305\u542b$x+y$\u7684\u9879\uff0c\u518d\u901a\u8fc7\u6362\u5143\u6cd5\u6c42\u89e3\u3002<br><br>$\\begin{split}\u89e3\uff1a\\\\&amp;\\because (x+y)^2=x^2+y^2+2xy\\\\&amp;\\therefore x^2+y^2=(x+y)^2-2xy=7\\ldots(1)\\\\&amp;\\because x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)(7-xy)=10\\ldots(2)\\\\&amp;\\therefore\u7531(2)\u5f0f\u53ef\u5f97\uff1axy=7-\\frac{10}{x+y}\\ldots(3)\\\\&amp;\u901a\u8fc7\u53d8\u6362\u540e\uff0c\u90fd\u53ea\u6709x+y\u9879\u4e86\uff0c\u4e0b\u4e00\u6b65\u5c31\u53ef\u4ee5\u8fdb\u884c\u6362\u5143\uff1a\u4ee4x+y=a\uff0c\u4ee3\u5165(1)(3)\u5f0f\u53ef\u5f97\uff1a\\\\&amp;a^2-2\\times\\left({7-\\frac{10}{a}}\\right)=7\\\\&amp;\\therefore a^2-14+\\frac{20}{a}-7=0\\\\&amp;\\therefore a^3-21a+20=0\\\\&amp;\\therefore a^3-1-(21a-21)=0\\\\&amp;(a-1)(a^2+a+1)-21(a-1)=0\\\\&amp;(a-1)(a^2+a-20)=0\\\\&amp;(a-1)(a+5)(a-4)=0\\\\&amp;\\therefore a_1=1\uff0ca_2=-5\uff0ca_3=4\\\\&amp;\\therefore x+y\u7684\u53d6\u503c\u5c31\u662f\u4e0a\u8ff0\u4e09\u4e2a\uff01\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-cb25745 stk-block-background\" data-block-id=\"cb25745\"><style>.stk-cb25745{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-cb25745:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-cb25745-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5a0f606\" data-v=\"4\" data-block-id=\"5a0f606\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-5a0f606-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-5a0f606-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-f00022e\" data-block-id=\"f00022e\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e8c\uff1a\u82e5$x^2-y^2=72\uff0c2xy+y^2=33$\uff0c\u6c42$x\u3001y$\u7684\u503c\u3002\u770b\u4f3c\u590d\u6742\uff0c\u5982\u679c\u7528\u5bf9\u65b9\u6cd5\uff0c\u5176\u5b9e\u4e5f\u4e0d\u96be\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u4e24\u4e2a\u6761\u4ef6\u5747\u4e3a\u4e8c\u6b21\u65b9\u7a0b\uff0c\u53ef\u4ee5\u8054\u7acb\u6c42\u89e3\uff0c\u4f46\u662f\u8fd9\u79cd\u8054\u7acb\u4e24\u4e2a\u65b9\u7a0b\u6c42\u89e3\u7684\u6b65\u9aa4\u8f83\u591a\uff0c\u5bb9\u6613\u51fa\u9519\u3002\u6b63\u786e\u7684\u65b9\u6cd5\u4f9d\u7136\u662f\u6362\u5143\u6cd5\uff0c\u5173\u952e\u662f\u5982\u4f55\u8fdb\u884c\u6362\u5143\u7684\u95ee\u9898\u3002\u901a\u8fc7\u89c2\u5bdf\u53d1\u73b0\uff0c\u4e24\u4e2a\u65b9\u7a0b\u5747\u6709\uff0c\u53ef\u4ee5\u8bbe\u6cd5\u6d88\u5143\u540e\u518d\u6362\u5143\u3002<br><br>$\\begin{split}\u89e3\uff1a\\\\&amp;\\because x^2-y^2=72\\ldots(1)\uff0c2xy+y^2=33\\ldots(2)\\\\&amp;\\therefore\u7531(2)\u5f0f\u53ef\u77e5y\\neq0\\Rightarrow y^2\\gt0\uff0c\u65b9\u7a0b\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5y^2\uff1a\\\\\\\\&amp;\\begin{cases}\\displaystyle\\frac{x^2}{y^2}-\\frac{y^2}{y^2}=\\frac{72}{y^2}\\\\ \\displaystyle\\frac{2xy}{y^2}+\\frac{y^2}{y^2}=\\frac{33}{y^2}\\end{cases}\\Rightarrow\\begin{cases}\\displaystyle\\left(\\frac{x}{y}\\right)^2-1=\\frac{72}{y^2}\\\\ \\displaystyle2\\frac{x}{y}+1=\\frac{33}{y^2}\\end{cases}\\\\\\\\&amp;\\ \u4ee4\\frac{x}{y}=a\u5219\u6709\uff1a\\begin{cases}\\displaystyle a^2-1=\\frac{72}{y^2}\\ldots(3)\\\\ \\displaystyle2a+1=\\frac{33}{y^2}\\ldots(4)\\end{cases}\\\\\\\\&amp;\u8054\u7acb(3)(4)\u5f0f\u53ef\u5f97\uff1a\\frac{a^2-1}{72}=\\frac{2a+1}{33}\\Rightarrow 33(a^2-1)=72(2a+1)\\\\&amp;33a^2-144a-105=0\\Rightarrow\\bbox[yellow,5pt,border:2px solid blue]{11a^2-48a-35=0}\\\\&amp;\\therefore (a-5)(11a+7)=0\\\\&amp;\\therefore a_1=5\uff0ca_2=-\\frac7{11}\\Rightarrow x=5y\u6216x=-\\frac7{11}y\\\\&amp;\u5206\u522b\u4ee3\u5165(1)\u5f0f\u53ef\u5f97\uff1a(5y)^2-y^2=72\\ldots(5)\uff0c(-\\frac7{11}y)^2-y^2=72\\ldots(6)\\\\&amp;\\therefore\u89e3\u5f97(5)\u5f0f\u7684\u6839\uff1ay=\\pm\\sqrt3\\ldots(7)\\\\&amp;\\because\u7531\u4e8ey^2\\gt0\\Rightarrow\\quad(6)\u5f0f\u5de6\u8fb9\uff1a\\left[(\\frac{7}{11})^2-1\\right]y^2\\lt0\\\\&amp;\\therefore(6)\u5f0f\u4e0d\u6210\u7acb\uff0cx=-\\frac{7}{11}y\u5e94\u820d\u53bb\u3002\\\\&amp;\u5c06(7)\u5f0f\u4ee3\u5165(1)\u5f0f\u53ef\u5f97\uff1ax^2-(\\pm\\sqrt{3})^2=72\\Rightarrow x^2=75\\Rightarrow x=\\pm5\\sqrt{3}\\\\&amp;\\because\u5c06(7)\u5f0f\u4ee3\u5165(2)\u5f0f\u53ef\u5f97\uff1a2xy=33-y^2=30\\gt0\\\\&amp;\\therefore x\uff0cy\u5fc5\u987b\u540c\u53f7\uff1a\\\\&amp;\\therefore\\bbox[yellow,8pt,border:2px solid red]{ \\begin{cases}x=5\\sqrt3\\\\y=\\sqrt3\\end{cases}\\text{\u6216}\\begin{cases}x=-5\\sqrt3\\\\y=-\\sqrt3\\end{cases}}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-ce901fb stk-block-background\" data-block-id=\"ce901fb\"><style>.stk-ce901fb{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-ce901fb:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-ce901fb-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-7b9a428\" data-v=\"4\" data-block-id=\"7b9a428\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-7b9a428-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-7b9a428-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-c1cddbe\" data-block-id=\"c1cddbe\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e09\uff1a\u82e5$a^2-a-1=0$\uff0c\u6c42$\\displaystyle a^8+\\frac{7}{a^4}$\u7684\u503c\u3002\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u4ee3\u6570\u5f0f\u6c42\u503c\uff0c\u6709\u4e00\u5b9a\u96be\u5ea6\uff0c\u9700\u8981\u6280\u5de7\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u6c42\u9ad8\u6b21\u5e42\u7684\u503c\uff0c\u901a\u5e38\u7684\u505a\u6cd5\u662f\u901a\u8fc7\u4ee3\u6570\u53d8\u5316\u8fdb\u884c\u964d\u6b21\uff0c\u6839\u636e\u5df2\u77e5\u6761\u4ef6\uff0c\u53ef\u4ee5\u8fde\u7eed\u8fdb\u884c\u964d\u5e42\uff0c\u7136\u540e\u6c42\u89e3\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\because a^2-a-1=0\\\\&amp;\\therefore a^2=a-1\\\\&amp;\\therefore a^8+\\frac{7}{a^4}=(a+1)^4+\\frac{7}{(a+1)^2}\\\\&amp;=(a^2+2a+1)^2+\\frac{7}{a^2+2a+1}\\\\&amp;=(3a+2)^2+\\frac{7}{3a+2}\\\\&amp;=9a^2+12a+4+\\frac{7}{3a+2}\\\\&amp;=21a+13+\\frac{7}{3a+2}\\\\&amp;=\\frac{(21a+13)(3a+2)+7}{3a+2}\\\\&amp;=\\frac{63a^2+81a+33}{3a+2}\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{=\\frac{144a+96}{3a+2}=\\frac{48(3a+2)}{3a+2}=48}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-a4e7738 stk-block-background\" data-block-id=\"a4e7738\"><style>.stk-a4e7738{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-a4e7738:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-a4e7738-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-67e956a\" data-v=\"4\" data-block-id=\"67e956a\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-67e956a-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-67e956a-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-3e590e9\" data-block-id=\"3e590e9\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u56db\uff1a\u82e5$x^2+y^2=29\uff0cxy+x+y=-13$\uff0c\u6c42$x\uff0cy$\u7684\u503c\u3002\u89e3\u6cd5\u4e0d\u62d8\u4e00\u683c\uff0c\u6709\u70b9\u610f\u601d\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u53ef\u4ee5\u901a\u8fc7\u8054\u7acb\u4e24\u4e2a\u65b9\u7a0b\u8fdb\u884c\u6c42\u89e3\uff1b\u4f46\u662f\uff0c\u8ba1\u7b97\u91cf\u6bd4\u8f83\u5927\uff0c\u5bb9\u6613\u51fa\u9519\u3002\u4ed4\u7ec6\u89c2\u5bdf\u53d1\u73b0\uff0c\u53ef\u4ee5\u901a\u8fc7\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\u8fdb\u884c\u964d\u6b21\uff0c\u7136\u540e\u518d\u6c42\u89e3\uff0c\u4e0d\u4ec5\u7b80\u5355\uff0c\u800c\u4e14\u4e0d\u6613\u51fa\u9519\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\begin{cases}x^2+y^2=29\\;\\qquad\\ldots(1)\\\\xy+x+y=-13\\ldots(2)\\end{cases}\\\\\\\\&amp;\\text{$(2)\u5f0f\\times2\u53ef\u5f97$\uff1a}2xy+2(x+y)=-26\\ldots(3)\\\\&amp;\\text{(1)+(3)\u53ef\u5f97\uff1a}x^2+y^2+2xy+2(x+y)=3\\\\\\\\&amp;(x+y)^2+2(x+y)-3=0\\\\&amp;\\therefore (x+y+3)(x+y-1)=0\\\\&amp;\\therefore x+y=-3\\text{\u6216\u8005}x+y=1\\\\\\\\&amp;\u5206\u522b\u4ee3\u5165(2)\u5f0f\u53ef\u5f97\uff1axy=-10\\text{\u6216\u8005}xy=-14\\\\\\\\&amp;\\therefore \\begin{cases}x+y=-3\\\\xy=-10\\end{cases}\\text{\u6216\u8005}\\begin{cases}x+y=1\\\\xy=-14\\end{cases}\\\\\\\\&amp;\\text{\u89e3\u4e0a\u8ff0\u4e24\u4e2a\u65b9\u7a0b\u7ec4\u53ef\u5f97\uff1a}\\\\\\\\&amp;\\therefore \\begin{cases}x_1=-5\\\\y_1=2\\end{cases}\\text{\u6216\u8005} \\begin{cases}x_2=2\\\\y_2=-5\\end{cases}\\\\\\\\&amp; \\begin{cases}x_3=\\displaystyle\\frac{1+\\sqrt{57}}{2}\\\\y_3=\\displaystyle\\frac{1-\\sqrt{57}}{2}\\end{cases}\\text{\u6216\u8005} \\begin{cases}x_4=\\displaystyle\\frac{1-\\sqrt{57}}{2}\\\\y_4=\\displaystyle\\frac{1+\\sqrt{57}}{2}\\end{cases}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-5b9c2d1 stk-block-background\" data-block-id=\"5b9c2d1\"><style>.stk-5b9c2d1{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-5b9c2d1:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-5b9c2d1-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2a6d720\" data-v=\"4\" data-block-id=\"2a6d720\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-2a6d720-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-2a6d720-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-ee660a4\" data-block-id=\"ee660a4\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e94\uff1a\u82e5$x^2=x+1\uff0cy^2=y+1$\uff0c\u6c42$x^2+y^2$\u7684\u503c\u3002\u4ee3\u6570\u5f0f\u6c42\u503c\uff0c\u539f\u6765\u8fd8\u6709\u8fd9\u79cd\u65b9\u6cd5\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u5982\u679c\u5148\u901a\u8fc7\u89e3\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u7684\u65b9\u6cd5\uff0c\u6c42\u51fa$x\u3001y$\u7684\u503c\uff0c\u7136\u540e\u4ee3\u5165\u6c42\u89e3\uff0c\u8fc7\u7a0b\u6bd4\u8f83\u5197\u957f\uff0c\u9700\u8981\u5206\u522b\u4ee3\u5165\u3002\u901a\u8fc7\u89c2\u5bdf\u53d1\u73b0\uff0c\u5df2\u77e5\u6761\u4ef6\u662f\u4e24\u4e2a\u4e00\u6a21\u4e00\u6837\u7684\u65b9\u7a0b\uff0c\u7531\u6b64\u8054\u60f3\u5230\u97e6\u8fbe\u5b9a\u7406\uff0c\u80fd\u5426\u7528\u8fd9\u4e2a\u5b9a\u7406\u6c42\u89e3\u3002<br><br>$\\begin{split} \\text{\u89e3\uff1a}\\\\&amp; \\text{\u6839\u636e\u5df2\u77e5\u6761\u4ef6\u53ef\u77e5\uff0c$x\u3001y$\u53ef\u4ee5\u770b\u4f5c\u65b9\u7a0b\uff1a}\\\\&amp;m^2=m+1\\Rightarrow m^2-m-1=0 \\text{\u7684\u4e24\u4e2a\u6839\uff0c\u6839\u636e\u97e6\u8fbe\u5b9a\u7406\uff1a} \\\\[2ex]&amp;\\because x+y=1\uff0cxy=-1\\\\\\\\&amp;\\therefore x^2+y^2=(x+y)^2-2xy\\\\&amp;=1^2-2\\times(-1)=3\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-b9e6233 stk-block-background\" data-block-id=\"b9e6233\"><style>.stk-b9e6233{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-b9e6233:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-b9e6233-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-67909ca\" data-v=\"4\" data-block-id=\"67909ca\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-67909ca-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-67909ca-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-426dff7\" data-block-id=\"426dff7\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u516d\uff1a\u82e5\u5b9e\u6570$x\u3001y$\u6ee1\u8db3$x^3+y^3=2$\uff0c\u6c42$x+y$\u7684\u6700\u5927\u503c\u30022017\u5e74\u9ad8\u8003\u6570\u5b66\u9898\uff0c\u4ee3\u6570\u5f0f\u6c42\u503c\uff0c\u7528\u521d\u4e2d\u77e5\u8bc6\u7167\u6837\u80fd\u591f\u89e3\u51fa\u6765\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u6761\u4ef6\u662f\u4e09\u6b21\u4ee3\u6570\u5f0f\uff0c\u8981\u6c42\u51fa$x\u3001y$\u7684\u6700\u5927\u503c\uff0c\u9700\u8981\u8fdb\u884c\u4ee3\u6570\u53d8\u6362\uff0c\u7136\u540e\u6784\u9020\u4e00\u4e2a\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\uff0c\u7528\u97e6\u8fbe\u5b9a\u7406\u6c42\u6700\u5927\u503c\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u6761\u4ef6\u4ee3\u6570\u5f0f\u53ef\u4ee5\u53d8\u6362\u4e3a\uff1a}(x+y)(x^2-xy+y^2)=2\\\\[2ex]&amp;\\therefore (x+y)\\left[(x+y)^2-3xy\\right]=2\\\\\\\\&amp;\\text{\u4ee4$x+y=t$(\u6839\u636e\u5df2\u77e5\u6761\u4ef6$t\\neq0)$}\\\\[2ex]&amp;\\therefore t(t^2-3xy)=2\\Rightarrow xy=\\frac13\\left(t^2-\\frac2t\\right)\\\\[2ex]\\\\&amp;\\therefore\\text{\u6839\u636e\u7ef4\u8fbe\u5b9a\u7406\uff0c$x\u3001y$\u662f\u4e0b\u9762\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u7684\u4e24\u4e2a\u6839\uff1a}\\\\&amp;M^2-tM+\\frac13(t^2-\\frac2t)=0\\\\&amp;\\therefore \\Delta=(-t)^2-4\\times1\\times\\frac13(t^2-\\frac2t)\\geqslant0\\\\&amp;\\text{\u6574\u7406\u53ef\u5f97\uff1a}t^2-\\frac8t\\leqslant0\\\\&amp;\\therefore t^2\\leqslant\\frac8t\\Rightarrow t^3\\leqslant8\\\\&amp;\\therefore t\\leqslant2\\\\&amp;\\therefore \\bbox[yellow,5pt,border:2px solid green]{(x+y)_{max}=2}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2003\u2003\u9898\u76ee\u4e00\uff1a\u82e5$x^2+y^2=7\uff0cx^3+y^3=10$\uff0c\u6c42$x+y$\u7684\u503c\u3002\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u96be\u5ea6\u592a\u5927\uff0c\u9700\u8981\u6280 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,4],"tags":[],"class_list":["post-497","post","type-post","status-publish","format-standard","hentry","category-3","category-4"],"_links":{"self":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/497","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=497"}],"version-history":[{"count":0,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/497\/revisions"}],"wp:attachment":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=497"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=497"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}