{"id":743,"date":"2024-02-21T17:23:52","date_gmt":"2024-02-21T09:23:52","guid":{"rendered":"https:\/\/big-tan.top\/?p=743"},"modified":"2024-02-22T12:48:56","modified_gmt":"2024-02-22T04:48:56","slug":"%e4%b8%ad%e5%ad%a6%e6%95%b0%e5%ad%a6%e5%8e%8b%e8%bd%b4%e9%a2%98%e9%9b%86%e9%94%a6%ef%bc%88%e4%b8%80%ef%bc%89","status":"publish","type":"post","link":"https:\/\/itantan.eu.org\/?p=743","title":{"rendered":"\u4e2d\u5b66\u6570\u5b66\u538b\u8f74\u9898\u96c6\u9526\uff08\u4e00\uff09"},"content":{"rendered":"\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-e6ccbbc stk-block-background\" data-block-id=\"e6ccbbc\"><style>.stk-e6ccbbc{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-e6ccbbc:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-e6ccbbc-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cffcf35\" data-v=\"4\" data-block-id=\"cffcf35\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-cffcf35-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-cffcf35-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-8a65ff9\" data-block-id=\"8a65ff9\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\u9898\u76ee\u4e00\uff1a\u89e3\u65b9\u7a0b (4+\\sqrt{15})^x+(4-\\sqrt{15})^x=62$\u3002\u590d\u65e6\u5927\u5b66\u9644\u4e2d\u81ea\u4e3b\u62db\u751f\u9898\u76ee\uff0c\u6709\u96be\u5ea6\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u89c2\u5bdf\u53d1\u73b0\u5e95\u6570\u4f3c\u4e4e\u6709\u4e9b\u76f8\u4f3c\uff0c\u5f97\u627e\u51fa\u6765\u8fd9\u4e24\u4e2a\u5e95\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\u624d\u80fd\u89e3\u51fa\u6765\u3002\u4f46\u662f\uff0c\u5982\u4f55\u627e\u51fa\u6765\u8fd9\u4e24\u4e2a\u5e95\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\u5462\uff1f\u4e0d\u59a8\u901a\u8fc7\u6362\u5143\u6cd5\u8bd5\u4e00\u4e0b\uff0c\u7c7b\u4f3c\u9898\u76ee\uff0c\u6362\u5143\u6cd5\u662f\u6bd4\u8f83\u5e38\u89c1\u7684\u5957\u8def\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u4ee4$(4+\\sqrt{15})^x=t$}\\\\&amp;\\because4-\\sqrt{15}=\\frac{(4-\\sqrt{15})(4+\\sqrt{15})}{4+\\sqrt{15}}=\\frac{4^2-(\\sqrt{15})^2}{4+\\sqrt{15}}=\\frac1{4+\\sqrt{15}}\\\\&amp;\\therefore (4-\\sqrt{15})^x=\\frac1t\\\\&amp;\\therefore \\text{\u539f\u65b9\u7a0b\u53ef\u4ee5\u8f6c\u6362\u4e3a\uff1a}t+\\frac1t=62\\\\&amp;\\therefore t^2-62t+1=0\\\\&amp;\\text{\u901a\u8fc7\u6c42\u6839\u516c\u5f0f\u53ef\u5f97\uff1a}t_1=31+8\\sqrt{15}\uff0ct_2=31-8\\sqrt{15}\\\\&amp;\\therefore (4+\\sqrt{15})^x=31+8\\sqrt{15}=(4+\\sqrt{15})^2\\Rightarrow x_1=2\\\\&amp;(4+\\sqrt{15})^x=31-8\\sqrt{15}=(4-\\sqrt{15})^2=(\\frac{1}{4+\\sqrt{15}})^2=(4+\\sqrt{15})^{-2}\\Rightarrow x_2=-2\\\\&amp;\\therefore x=\\pm2\\end{split}$<\/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-508260b stk-block-background\" data-block-id=\"508260b\"><style>.stk-508260b{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-508260b:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-508260b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5508206\" data-v=\"4\" data-block-id=\"5508206\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-5508206-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-5508206-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-6e3926b\" data-block-id=\"6e3926b\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\u9898\u76ee\u4e8c\uff1a\\displaystyle\u5df2\u77e5\uff1aa=\\sqrt[3]4+\\sqrt[3]2+1\uff0c\u6c42\uff1a\\frac3a+\\frac{3}{a^2}+\\frac1{a^3}\u7684\u503c\u3002\u4e2d\u8003\u6570\u5b66\u538b\u8f74\u9898\uff0c\u5927\u591a\u6570\u4eba\u675f\u624b\u65e0\u7b56\uff01$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u4f3c\u4e4e\u65e0\u4ece\u4e0b\u624b\uff0c\u4f46\u662f\uff0c\u901a\u8fc7\u89c2\u5bdf\u53d1\u73b0\uff0c\u628a\u9700\u8981\u8ba1\u7b97\u7684\u7b97\u5f0f\u8fdb\u884c\u901a\u5206\u4e4b\u540e\uff0c\u4f3c\u4e4e\u8ddf\u7acb\u65b9\u5dee\u7684\u516c\u5f0f\u6709\u4e9b\u8054\u7cfb\uff0c\u90a3\u4e0d\u59a8\u4ece\u8fd9\u91cc\u8003\u8651\u8ba1\u7b97\u4ee3\u6570\u5f0f\u7684\u503c\uff0c\u5f53\u7136\uff0c\u6362\u5143\u6cd5\u8fd8\u662f\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u624b\u6bb5\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\because\\frac3a+\\frac{3}{a^2}+\\frac1{a^3}=\\frac{3a^2+3a+1}{a^3}=\\frac{(a+1)^3-a^3}{a^3}=\\left(\\frac{a+1}{a}\\right)^3-1\\\\&amp;\\text{\u4ee4$m=\\sqrt[3]2$}\\text{\uff0c\u5219\uff1a}a=m^2+m+1\\\\&amp;(m-1)\\cdot a=(m-1)(m^2+m+1)=m^3-1^3=\\left(\\sqrt[3]2\\right)^3-1=1\\\\&amp;\\therefore a=\\frac1{m-1}\\\\&amp;\\therefore\\displaystyle \\frac{a+1}{a}=\\frac{\\frac1{m-1}+1}{\\frac{1}{m-1}}=m\\\\&amp;\\therefore \\frac3a+\\frac{3}{a^2}+\\frac1{a^3}=m^3-1=(\\sqrt[3]2)^3-1=1\\end{split}$<\/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-f9f9a2d stk-block-background\" data-block-id=\"f9f9a2d\"><style>.stk-f9f9a2d{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-f9f9a2d:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-f9f9a2d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3f7b4d5\" data-v=\"4\" data-block-id=\"3f7b4d5\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-3f7b4d5-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-3f7b4d5-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-097e622\" data-block-id=\"097e622\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\u9898\u76ee\u4e09\uff1a\u89e3\u65b9\u7a0b \\sqrt{7x^2+9x+13}+\\sqrt{7x^2-5x+13}=7x$\uff0c1995\u5e74\u67d0\u5730\u65b9\u4e2d\u8003\u6570\u5b66\u538b\u8f74\u9898\uff0c\u96be\u5ea6\u8f83\u5927\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7c7b\u4f3c\u9898\u76ee\uff0c\u6700\u5bb9\u6613\u60f3\u5230\u7684\u5c31\u662f\u4f7f\u7528\u6362\u5143\u6cd5\uff0c\u5e38\u89c1\u7684\u6362\u5143\u65b9\u5f0f\u6709\u76f4\u63a5\u6362\u5143\u3001\u5747\u503c\u6362\u5143\uff0c\u4f46\u662f\uff0c\u8fd9\u9053\u9898\u76ee\u4f3c\u4e4e\u90fd\u65e0\u6cd5\u7528\u4e00\u822c\u7684\u65b9\u5f0f\u8fdb\u884c\u6362\u5143\uff0c\u600e\u95e8\u529e\uff1f\u90a3\u5c31\u7ed3\u5408\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u8fdb\u884c\u7efc\u5408\u6362\u5143\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u4ee4$a=7x^2+2x+13\uff0cb=7x$}\\\\&amp;a\\pm b\\geqslant0\uff0cb\\geqslant0\\\\&amp;\\text{\u5219\u539f\u65b9\u7a0b\u53ef\u4ee5\u8f6c\u6362\u4e3a\uff1a}\\sqrt{a+b}+\\sqrt{a-b}=b\\\\&amp;\\therefore (\\sqrt{a+b}+\\sqrt{a-b})^2=b^2\\\\&amp;\\therefore 2a+2\\sqrt{a^2-b^2}=b^2\\\\&amp;\\therefore 2\\sqrt{a^2-b^2}=b^2-2a\\\\&amp;\\therefore 4a^2-4b^2=b^4-4ab^2+4a^2\\\\&amp;\\therefore b^2=4a-4\\\\&amp;\\text{\u8fd8\u539f\u540e\u53ef\u5f97\uff1a}(7x)^2=4(7x^2+2x+13)-4\\Rightarrow 21x^2-8x-48=0\\\\&amp;\\therefore(7x-12)(3x+4)=0\\\\&amp;\\text{\u89e3\u5f97\uff1a}x_1=\\frac{12}7\uff0cx_2=-\\frac43\\\\&amp;\\because b\\geqslant0\\\\&amp;\\therefore x=\\frac{12}7\\end{split}$<\/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-29bed04 stk-block-background\" data-block-id=\"29bed04\"><style>.stk-29bed04{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-29bed04:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-29bed04-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9d01b2e\" data-v=\"4\" data-block-id=\"9d01b2e\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-9d01b2e-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-9d01b2e-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-af8d6c4\" data-block-id=\"af8d6c4\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\\begin{split}&amp;\u9898\u76ee\u56db\uff1a\u6b63\u6570 a\u3001b\u3001c  \u6ee1\u8db3 a^{\\frac32}+b^{\\frac32}+c^{\\frac32}=1\uff0c\\\\&amp;\u6c42\u8bc1\uff1a\\\\&amp;(1)abc\\leqslant\\frac19\uff0c\\\\&amp;(2)\\frac{c}{a+b}+\\frac{a}{b+c}+\\frac{b}{c+a}\\leqslant\\frac{1}{2\\sqrt{abc}}\\\\\\\\&amp;2022\u5e74\u9ad8\u8003\u6587\u79d1\u538b\u8f74\u9898\uff0c\u7528\u521d\u4e2d\u6570\u5b66\u77e5\u8bc6\u5c45\u7136\u4e5f\u53ef\u4ee5\u6c42\u89e3!\\end{split}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u4f3c\u4e4e\u96be\u5ea6\u8f83\u5927\uff0c\u4f46\u662f\uff0c\u53ea\u8981\u719f\u7ec3\u638c\u63e1\u6362\u5143\u6cd5\u4ee5\u53ca\u4ee3\u6570\u5f0f\u8f6c\u6362\u7684\u6280\u5de7\uff0c\u6c42\u89e3\u8d77\u6765\u7684\u96be\u5ea6\u5e76\u4e0d\u662f\u592a\u5927\uff0c\u5145\u5176\u91cf\u53ea\u662f\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\u7684\u96be\u5ea6\uff01<br><br>$\\begin{split}\u8bc1\u660e\uff1a(\u4e00)\\\\&amp;\\because\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\uff1a\\frac{a+b+c}{3}\\geqslant \\sqrt[3]{abc}\\\\&amp;\\therefore 1= a^{\\frac32}+b^{\\frac32}+c^{\\frac32}\\geqslant 3\\sqrt[3]{a^\\frac32\\cdot b^\\frac32\\cdot c^\\frac32}=3\\sqrt{abc}\\\\&amp;\\therefore \\sqrt{abc}\\leqslant\\frac13\\Rightarrow abc\\leqslant\\frac19\\\\&amp;\\therefore\u7b2c\u4e00\u95ee\u7684\u96be\u5ea6\u5176\u5b9e\u5e76\u4e0d\u5927\uff0c\u91cd\u70b9\u5c31\u662f\u8003\u5bdf\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\u8fd9\u4e2a\u77e5\u8bc6\u70b9\u7f62\u4e86\u3002\\end{split}$<br><br>$\\begin{split}\u8bc1\u660e\uff1a(\u4e8c)\\\\&amp;\u8bbea=m^2\uff0cb=n^2\uff0cc=t^2\uff0c(m\u3001n\u3001t\u5747\u5927\u4e8e0)\\\\&amp;\u5219\u7b2c\u4e8c\u95ee\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8bc1\u660e\uff1a\\frac{t^2}{m^2+n^2}+\\frac{m^2}{n^2+t^2}+\\frac{n^2}{m^2+t^2}\\leqslant \\frac1{2mnt}\\ldots(1)\\\\&amp;\\because m^3+n^3+t^3=1\uff0ca\u3001b\u3001c \u5747\u4e3a\u6b63\u6570\\\\&amp;\\therefore(m-n)^2\\geqslant0\\Rightarrow m^2+n^2\\geqslant2mn\\Rightarrow \\frac{1}{m^2+n^2}\\leqslant\\frac1{2mn}\\\\&amp;\\therefore \\frac{1}{n^2+t^2}\\leqslant\\frac1{2nt}\uff0c\\frac{1}{m^2+t^2}\\leqslant\\frac1{2mt}\\\\&amp;\\therefore (1)\u53ef\u8f6c\u6362\u4e3a\uff1a\\frac{t^2t}{(m^2+n^2)t}+\\frac{m^2m}{(n^2+t^2)m}+\\frac{n^2n}{(m^2+t^2)n}\\leqslant \\frac1{2mnt}\\cdots(2)\\\\&amp;\\therefore(2)\u5de6\u8fb9\\leqslant \\frac{t^3}{2mn\\cdot t}+\\frac{m^3}{2nt\\cdot m}+\\frac{n^3}{2mt\\cdot n}=\\frac{t^3+m^3+n^3}{2mnt}=\\frac1{2mnt}\\\\&amp;\\therefore \\frac{c}{a+b}+\\frac{a}{b+c}+\\frac{b}{c+a}\\leqslant\\frac1{2\\sqrt{abc}}\\\\\\\\&amp;\\therefore \u770b\u8d77\u6765\u662f\u9ad8\u8003\u6587\u79d1\u7684\u538b\u8f74\u9898\u76ee\uff0c\u5176\u5b9e\u4e5f\u5c31\u662f\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u6c34\u5e73\uff01\uff01\\end{split}$<\/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-a5922a3 stk-block-background\" data-block-id=\"a5922a3\"><style>.stk-a5922a3{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-a5922a3:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-a5922a3-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5cf5d95\" data-v=\"4\" data-block-id=\"5cf5d95\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-5cf5d95-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-5cf5d95-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-70d92fa\" data-block-id=\"70d92fa\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e94\uff1a\u82e5$x^2+y^2=9$\uff0c\u6c42$\\sqrt{x^2-6x+9}+\\sqrt{xy-4x+3y-12}$\u7684\u503c\u3002\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u538b\u8f74\u9898\uff0c\u5b66\u9738\u4e5f\u89c9\u5f97\u65e0\u4ece\u4e0b\u624b\uff01<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u770b\u8d77\u6765\u8fd9\u9053\u9898\u76ee\u4f3c\u4e4e\u5e76\u4e0d\u662f\u5f88\u96be\uff0c\u4ee3\u6570\u5f0f\u4e5f\u5f88\u5bb9\u6613\u5316\u7b80\uff0c\u4f46\u662f\uff0c\u5316\u7b80\u4ee3\u6570\u5f0f\u4e4b\u540e\uff0c\u5982\u4f55\u6c42\u503c\u5462\uff1f\u4e0d\u662f\u592a\u7b80\u5355\uff0c\u8fd9\u9053\u9898\u76ee\u7684\u5173\u952e\u662f\u5982\u4f55\u5145\u5206\u5229\u7528\u5df2\u77e5\u6761\u4ef6\uff0c\u901a\u8fc7\u5df2\u77e5\u6761\u4ef6\u6765\u6316\u6398\u5176\u4e2d\u7684\u6df1\u9083\u610f\u4e49\uff0c\u5982\u679c\u65e0\u6cd5\u5145\u5206\u5229\u7528\u5df2\u77e5\u6761\u4ef6\uff0c\u5b8c\u6574\u6c42\u89e3\u4f3c\u4e4e\u4e0d\u662f\u90a3\u4e48\u5bb9\u6613\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\u539f\u5f0f=\\sqrt{(x-3)^2}+\\sqrt{(y-4)(x+3)}=\\vert x-3\\vert+\\sqrt{(x+3)(y-4)}\\\\&amp;\\therefore \\text{\u5230\u8fd9\u4e00\u6b65\u4f3c\u4e4e\u4e0d\u96be\uff0c\u4f46\u662f\u63a5\u4e0b\u6765\uff0c\u5982\u4f55\u6c42\u503c\u5462\uff1f\u4f3c\u4e4e\u65e0\u4ece\u4e0b\u624b\u4e86\uff01}\\\\&amp;\\because x^2+y^2=9\\\\&amp;\\therefore x^2=9-y^2\\geqslant0\\Rightarrow y^2\\leqslant9\\Rightarrow -3\\leqslant y\\leqslant3\\\\&amp;\\text{\u540c\u7406\uff1a$-3\\leqslant x\\leqslant3$}\\\\&amp;\\therefore y-4\\lt0\uff0cx+3\\geqslant0\\\\&amp;\\because (x+3)(y-4)\\geqslant0\\\\&amp;\\therefore (x+3)(y-4)=0\uff0c\u53ea\u80fd\u7b49\u4e8e0\u624d\u80fd\u7b26\u5408\u6761\u4ef6\uff01\\\\&amp;x=-3\\\\&amp;\\bbox[yellow,5pt,border:2px solid red]{\u539f\u5f0f=6}\\\\&amp;\\bbox[yellow,5pt,border:2px solid blue]{\\text{\u53e6\u5916\uff0c\u5982\u679c\u5b66\u8fc7\u9ad8\u4e2d\u89e3\u6790\u51e0\u4f55\uff0c$x^2+y^2=9$\u5176\u5b9e\u5c31\u662f\u4e00\u4e2a\u5706\u7684\u65b9\u7a0b\uff01}}\\end{split}$<\/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-6b0f414 stk-block-background\" data-block-id=\"6b0f414\"><style>.stk-6b0f414{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-6b0f414:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-6b0f414-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-6a0cdb6\" data-v=\"4\" data-block-id=\"6a0cdb6\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-6a0cdb6-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-6a0cdb6-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-fda4a1c\" data-block-id=\"fda4a1c\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u516d\uff1a\u5df2\u77e5$x\u3001y$\u5747\u4e3a\u6574\u6570\uff0c$15x^2y^2=35x^2-3y^2+412$\uff0c\u6c42$xy$\u7684\u503c\u3002\u6570\u5b66\u7ade\u8d5b\u538b\u8f74\u9898\uff0c\u9700\u8981\u6709\u7f1c\u5bc6\u7684\u601d\u8def\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7531\u4e8e\u6709\u4e24\u4e2a\u53d8\u91cf\uff0c\u65e0\u6cd5\u76f4\u63a5\u628a$x\u3001y$\u89e3\u51fa\u6765\uff0c\u53ea\u80fd\u53e6\u8f9f\u8e4a\u5f84\u4e86\uff0c\u7c7b\u4f3c\u9898\u76ee\u53ef\u4ee5\u8bbe\u6cd5\u5c06\u6761\u4ef6\u7b49\u5f0f\u5de6\u8fb9\u901a\u8fc7\u56e0\u5f0f\u5206\u89e3\u53d8\u6210\u4e24\u4e2a\u4ee3\u6570\u5f0f\u76f8\u4e58\uff0c\u7b49\u4e8e\u4e00\u4e2a\u5e38\u6570\u7684\u5f62\u5f0f\uff0c\u7136\u540e\u5229\u7528$x\u3001y$\u5747\u4e3a\u6574\u6570\u7684\u6761\u4ef6\u8fdb\u884c\u5224\u65ad\uff0c\u7136\u540e\u6c42\u51fa$x\u548cy$\u7684\u4e58\u79ef\u503c\u3002<br><br>$\\begin{split}\u89e3\uff1a\\\\&amp;\\text{\u7531\u5df2\u77e5\u6761\u4ef6\u53ef\u77e5\uff1a}5x^2(3y^2-7)+3y^2-7=412-7\\\\&amp;(5x^2+1)(3y^2-7)=405=3^4\\cdot5\uff0c\u5173\u952e\u662f\u9700\u8981\u628a405\u8fd9\u4e2a\u6570\u5206\u89e3\u6210\u51e0\u4e2a\u8d28\u6570\u7684\u4e58\u79ef\uff01\u7136\u540e\u5224\u65ad\u6c42\u89e3\uff01\\\\&amp;\u6839\u636e\u5df2\u77e5\u6761\u4ef6\u53ca\u6574\u6570\u7684\u6027\u8d28\uff1a\\\\&amp;5x^2+1\u4e0d\u80fd\u88ab5\u6574\u9664\uff1b3y^2-7\u4e0d\u80fd\u88ab3\u6574\u9664(\u8fd9\u4e00\u6b65\u5229\u7528\u4e86\u6574\u6570\u7684\u6027\u8d28\u4ee5\u53ca\u6574\u6570\u8868\u793a\u6cd5\u6765\u5224\u65ad\u6574\u9664\u6027)\\\\&amp;\\therefore \\begin{cases}5x^2+1\\gt0\\\\3y^2-7\\gt0\\end{cases}\\Rightarrow \\begin{cases}5x^2+1=3^4\\\\3y^2-7=5\\end{cases}\\Rightarrow \\begin{cases}5x^2=80\\\\3y^2=12\\end{cases}\\\\&amp;\\therefore 5x^2\\cdot3y^2=960\\Rightarrow x^2y^2=64\\\\&amp;\\therefore xy=\\pm8\\end{split}$<br><br><br>\u2003\u2003<strong>\u62d3\u5c55\u601d\u7ef4\u9898\uff1a$\\bbox[yellow,5pt,border:2px solid red]{\u6c42\u6307\u6570\u65b9\u7a0b\u7684\u6b63\u6574\u6570\u89e3\uff1a4^x-36^y=28}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7c7b\u4f3c\u65b9\u7a0b\u6709\u4e24\u4e2a\u672a\u77e5\u6570\uff0c\u5fc5\u987b\u901a\u8fc7\u5df2\u77e5\u6761\u4ef6\u8fdb\u884c\u6c42\u89e3\uff0c\u6839\u636e\u6574\u6570\u7684\u6027\u8d28\uff1a\u6574\u6570\u7684\u6574\u6570\u6b21\u65b9\u4e00\u5b9a\u8fd8\u662f\u6574\u6570\uff1b\u5076\u6570\u7684\u6574\u6570\u6b21\u65b9\u8fd8\u662f\u5076\u6570\uff1b\u5947\u6570\u7684\u6574\u6570\u6b21\u65b9\u8fd8\u662f\u5947\u6570\uff01\u4e24\u4e2a\u5947\u6570\u7684\u4e58\u79ef\u8fd8\u662f\u5947\u6570\uff01<br><br>$\\begin{split}\u89e3\uff1a\\\\&amp;\u539f\u65b9\u7a0b\u53ef\u4ee5\u53d8\u6362\u4e3a\uff1a(2^x)^2-(6^y)^2=28\\\\&amp;(2^x+6^y)(2^x-6^y)=28\\\\&amp;\\because \u6839\u636e\u5df2\u77e5\u6761\u4ef6\u4ee5\u53ca\u6574\u6570\u7684\u6027\u8d28\uff0c2^x\\pm6^y\u5747\u4e3a\u6b63\u5076\u6570\u4e142^x+6^y\\gt2^x-6^y\\\\&amp;\\because\u7531\u4e8e28\u552f\u4e00\u53ef\u4ee5\u5206\u89e3\u6210\u4e24\u4e2a\u5076\u6570\u4e58\u79ef\u7684\u65b9\u5f0f\u4e3a2\\times14\\\\&amp;\\therefore \\begin{cases}2^x+6^y=14\\\\\\\\2^x-6^y=2\\end{cases}\\Rightarrow\\bbox[yellow,5pt,border:2px solid red]{x=3\uff0cy=1}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2003\u2003$\u9898\u76ee\u4e00\uff1a\u89e3\u65b9\u7a0b (4+\\sqrt{15})^x+(4-\\sqrt{15})^x=62$\u3002\u590d\u65e6\u5927\u5b66\u9644\u4e2d\u81ea\u4e3b [&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-743","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\/743","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=743"}],"version-history":[{"count":0,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/743\/revisions"}],"wp:attachment":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=743"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=743"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}