{"id":21395,"date":"2026-02-04T14:24:06","date_gmt":"2026-02-04T05:24:06","guid":{"rendered":"https:\/\/math-travel.jp\/?p=21395"},"modified":"2026-03-06T01:06:59","modified_gmt":"2026-03-05T16:06:59","slug":"solution-formula","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-1\/solution-formula\/","title":{"rendered":"\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u516c\u5f0f\u3068\u306f\uff1f\u899a\u3048\u65b9\u3068\u8a08\u7b97\u3092\u30e9\u30af\u306b\u3059\u308b\u4f7f\u3044\u65b9\u3092\u4f8b\u984c\u3067\u30de\u30b9\u30bf\u30fc"},"content":{"rendered":"\n

\u6570\u5b66\u2160\u4e8c\u6b21\u95a2\u6570\u306b\u306f\u300c\u89e3\u306e\u516c\u5f0f<\/span>\u300d\u3092\u6d3b\u7528\u3059\u308b\u554f\u984c\u304c\u591a\u304f\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\n

\u300c\u89e3\u306e\u516c\u5f0f\u3092\u5fd8\u308c\u3066\u3057\u307e\u3063\u305f\u300d<\/span>
\u300c\u89e3\u306e\u516c\u5f0f\u304c\u4f7f\u3044\u3053\u306a\u305b\u306a\u3044\u300d<\/span><\/p>\n<\/div><\/div>\n\n\n\n

\u4eca\u56de\u306f\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u516c\u5f0f\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u89e3\u306e\u516c\u5f0f\"
\u89e3\u306e\u516c\u5f0f<\/figcaption><\/figure>\n<\/div>\n\n\n

\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u30921\u767a\u3067\u6c42\u3081\u3089\u308c\u308b\u516c\u5f0f\u304c\u89e3\u306e\u516c\u5f0f\u3067\u3059\u3002
\u672c\u8a18\u4e8b\u3067\u306f\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u3092\u89e3\u8aac<\/span>\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u305d\u3082\u305d\u3082”\u89e3”\u3068\u306f\u4f55\u306a\u306e\u304b\u3001\u305d\u3046\u3044\u3063\u305f\u57fa\u790e\u304b\u3089\u89e3\u8aac\u3057\u3066\u3044\u308b\u306e\u3067\u305c\u3072\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

2\u6b21\u95a2\u6570\u3068\u306f<\/h2>\n\n\n\n

2\u6b21\u95a2\u6570\u3068\u306f\u6700\u5927\u6b21\u6570\u304c2\u306e\u95a2\u6570\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n

\n
\"2\u6b21\u95a2\u6570\"<\/figure>\n<\/div>\n\n\n
\n

\u25a0 2\u6b21\u95a2\u6570\u306e\u4f8b<\/p>\n\n\n\n

\\\\(y=x^{2}+3x+4\\)
\\(y=x^{2}\\)
\\(y=x^{2}-5\\)<\/p>\n<\/div><\/div>\n\n\n\n

\u95a2\u6570\u3068\u306f\uff1f<\/span><\/div>
\n

\\(x\\)\u306e\u5024\u30921\u3064\u6c7a\u3081\u305f\u3068\u304d\u3001\u305d\u308c\u306b\u4f34\u3063\u3066\\(y\\)\u306e\u5024\u30821\u3064\u306b\u6c7a\u307e\u308b\u6570\u5f0f\u306e\u3053\u3068\u3002<\/p>\n<\/div><\/div>\n\n\n\n

2\u6b21\u95a2\u6570\u3092\u7dcf\u5fa9\u7fd2\u3057\u305f\u3044\u65b9\u306f\u3053\u3061\u3089\u306e\u8a18\u4e8b\u304c\u304a\u3059\u3059\u3081\u3067\u3059\u3002<\/p>\n\n\n\n

\uff1e\uff1e2\u6b21\u95a2\u6570\u306e\u516c\u5f0f\u3068\u91cd\u8981\u30dd\u30a4\u30f3\u30c8\u307e\u3068\u3081<\/p>\n\n\n\n

\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3068\u89e3<\/h2>\n\n\n\n
\"\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3068\u89e3\"<\/figure>\n\n\n\n

\u305d\u3057\u3066\u30012\u6b21\u95a2\u6570\u306e\u65b9\u7a0b\u5f0f\u3092\u4e8c\u6b21\u65b9\u7a0b\u5f0f<\/span>\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u4e8c\u6b21\u65b9\u7a0b\u5f0f\"<\/figure>\n<\/div>\n\n\n

\u4ee3\u8868\u7684\u306a\u5f62\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f0f\u3067\u3059\u3002<\/p>\n\n\n\n

\\[ax^{2}+bx+c=0\\]<\/p>\n\n\n\n

\\(a,b,c\\)\u306f\u4fc2\u6570\u3067\u3001\\(a \\neq 0\\)\u3067\u3059\u3002\uff08\\(a=0\\)\u3060\u30681\u6b21\u5f0f\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u305f\u3081\u3002\uff09<\/p>\n\n\n\n

\n

\u25a0 2\u6b21\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/p>\n\n\n\n

\\(x^{2}+5x+4=0\\)
<\/strong>\\(-2^{2}+3x+1=0\\)<\/p>\n<\/div><\/div>\n\n\n\n

\u53c2\u8003<\/span><\/div>
\n

\\(x^{3}+3x^{2}+x+4=0\\)\u306f3\u6b21\u65b9\u7a0b\u5f0f
\\(x^{4}+5x^{3}+2x^{2}+x+4=0\\)\u306f4\u6b21\u65b9\u7a0b\u5f0f<\/p>\n<\/div><\/div>\n\n\n\n

\u307e\u305f\u3001\u4e8c\u6b21\u65b9\u7a0b\u5f0f\\(ax^{2}+bx+c=0\\)\u3092\u6210\u308a\u7acb\u305f\u305b\u308b\\(x\\)\u306e\u5024\u3092\u65b9\u7a0b\u5f0f\u306e\u89e3<\/span>\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u5177\u4f53\u7684\u306b\u8a00\u3046\u3068\u3001\\(x^{2}-5x+4=0\\)\u306e\u89e3\u306f\\(x=1,4\\)\u3067\u3059\u3002<\/p>\n\n\n\n

\\(x=1\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001
\\[1^{2}-5 \\cdot 1+4=0\\]<\/p>\n\n\n\n

\\(x=4\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001
\\[4^{2}-5 \\cdot 4+4=0\\]<\/p>\n\n\n\n

\u78ba\u304b\u306b\\(x=1,4\\)\u306e\u3068\u304d\u306b\u65b9\u7a0b\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u3061\u306a\u307f\u306b\u3001\\(x^{2}-5x+4=0\\)\u306e\u89e3\u3068\u3044\u3046\u306e\u306f\\(y=x^{2}-5x+4\\)\u306e\u30b0\u30e9\u30d5\u3068x\u8ef8\u3068\u306e\u4ea4\u70b9\u3092\u6307\u3057\u307e\u3059\u3002
\u3064\u307e\u308a\u3001\\(x=1,4\\)\u304c\u65b9\u7a0b\u5f0f\u306e\u89e3\u306a\u3089\u3070\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u65b9\u7a0b\u5f0f\u306e\u89e3\u304c\u8868\u3059\u3082\u306e\"<\/figure>\n<\/div>\n\n\n

\u89e3\u306e\u516c\u5f0f\u3068\u4e8c\u6b21\u65b9\u7a0b\u5f0f<\/h2>\n\n\n
\n
\"\u89e3\u306e\u516c\u5f0f\u3068\u4e8c\u6b21\u65b9\u7a0b\u5f0f\"<\/figure>\n<\/div>\n\n\n

\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u6c42\u3081\u65b9\u306f\u4e3b\u306b2\u3064\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u6c42\u3081\u65b9<\/span><\/div>
\n
    \n
  • \u56e0\u6570\u5206\u89e3\u3057\u3066\u6c42\u3081\u308b<\/li>\n\n\n\n
  • \u89e3\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u6c42\u3081\u308b<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

    \u4e0b\u306e\u6570\u5f0f\u304c\u89e3\u306e\u516c\u5f0f\u3067\u3059\u3002<\/p>\n\n\n

    \n
    \"\u89e3\u306e\u516c\u5f0f\"<\/figure>\n<\/div>\n\n\n

    \u89e3\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3068\u3069\u3093\u306a\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3067\u3082\u3001\u7c21\u5358\u306b\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

    \u56e0\u6570\u5206\u89e3\u3067\u89e3\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n

    \u89e3\u306e\u516c\u5f0f\u3092\u89e3\u8aac\u3059\u308b\u524d\u306b\u56e0\u6570\u5206\u89e3\u3067\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u5fc5\u305a\u62bc\u3055\u3048\u3066\u6b32\u3057\u3044\u306e\u3067\u78ba\u8a8d\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    x^{2}-5x+4&=&0\\\\
    (x-1)(x-4)&=&0
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\\(x=1,4\\)<\/p>\n\n\n\n

    \\begin{eqnarray}
    2x^{2}-4x-6&=&0\\\\
    2(x^{2}-2x-3)&=&0\\\\
    2(x+1)(x-3)&=&0
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\\(x=-1,3\\)<\/p>\n\n\n\n

    \u89e3\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u6c42\u3081\u308b<\/h3>\n\n\n\n
    \u89e3\u306e\u516c\u5f0f<\/span><\/div>
    \n

    \"\u89e3\u306e\u516c\u5f0f\"<\/p>\n<\/div><\/div>\n\n\n\n

    \u304b\u306a\u308a\u4fbf\u5229\u306a\u516c\u5f0f\u306a\u306e\u3067\u89e3\u306e\u516c\u5f0f\u306f\u5fc5\u305a\u899a\u3048\u307e\u3057\u3087\u3046\u3002<\/span><\/p>\n\n\n\n

    \u306a\u305c\u306a\u3089\u3001\u516c\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3060\u3051\u3067\u3069\u3093\u306a\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u3082\u6c42\u3081\u3089\u308c\u308b<\/span>\u304b\u3089\u3067\u3059\u3002<\/p>\n\n\n\n

    \u4e0b\u306e\u65b9\u7a0b\u5f0f\u3088\u3046\u306a\u30ad\u30ec\u30a4\u306b\u56e0\u6570\u5206\u89e3\u304c\u3067\u304d\u306a\u3044\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3067\u3082\u7c21\u5358\u306b\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

    \\(x^{2}+4x-2=0\\)<\/p>\n\n\n\n

    \\(a=1,b=4,c=-2\\)\u3068\u3057\u3066\u3001\u89e3\u306e\u516c\u5f0f\u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\frac{-b\u00b1\\sqrt{b^{2}-4ac}}{2a}&=&\\frac{-4\u00b1\\sqrt{4^{2}-4 \\cdot 1 \\cdot (-2)}}{2 \\cdot 1}\\\\
    &=&\\frac{-4\u00b1\\sqrt{16+8}}{2}\\\\
    &=&\\frac{-4\u00b1\\sqrt{24}}{2}\\\\
    &=&-2\u00b1\\sqrt{6}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n

    \\(x=-2\u00b1\\sqrt{6}\\)<\/p>\n\n\n\n

    \u89e3\u306e\u516c\u5f0f\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n

    \u89e3\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u3092\u6c42\u3081\u308b\u7df4\u7fd2\u3092\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \u7df4\u7fd2\u554f\u984c<\/span><\/div>
    \n

    \u6b21\u306e\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u3092\u6c42\u3081\u3088\u3046\u3002<\/p>\n\n\n\n

    (1)\u3000\\(x^{2}-x-3=0\\)
    (2)\u3000\\(-2x^{2}+x-3=0\\)<\/p>\n<\/div><\/div>\n\n\n\n

    \\(x^{2}-x-3=0\\)<\/h3>\n\n\n\n

    \\(a=1,b=-1,c=-3\\)\u3068\u3057\u3066\u3001\u89e3\u306e\u516c\u5f0f\u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\frac{-b\u00b1\\sqrt{b^{2}-4ac}}{2a}&=&\\frac{-(-1)\u00b1\\sqrt{(-1)^{2}-4 \\cdot 1 \\cdot (-3)}}{2 \\cdot 1}\\\\
    &=&\\frac{1\u00b1\\sqrt{1+12}}{2}\\\\
    &=&\\frac{1\u00b1\\sqrt{13}}{2}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n

    \\[\\displaystyle x=\\frac{1\u00b1\\sqrt{13}}{2}\\]<\/p>\n\n\n\n

    \\(-2x^{2}+x+3=0\\)<\/h3>\n\n\n\n

    \\(a=-2,b=1,c=-3\\)\u3068\u3057\u3066\u3001\u89e3\u306e\u516c\u5f0f\u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\frac{-b\u00b1\\sqrt{b^{2}-4ac}}{2a}&=&\\frac{-1\u00b1\\sqrt{1^{2}-4 \\cdot (-2) \\cdot 3}}{2 \\cdot (-2)}\\\\
    &=&\\frac{-1\u00b1\\sqrt{1+24}}{-4}\\\\
    &=&\\frac{1\u00b15}{4}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n

    \\[\\displaystyle x=-1,\\frac{3}{2}\\]<\/p>\n\n\n\n

    \u89e3\u306e\u516c\u5f0f \u307e\u3068\u3081<\/h2>\n\n\n\n

    \u4eca\u56de\u306f\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u516c\u5f0f\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n

    \n
    \"\u89e3\u306e\u516c\u5f0f\"<\/figure>\n<\/div>\n\n\n

    \u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u6c42\u3081\u65b9\u306f\u4e3b\u306b2\u3064<\/span><\/p>\n\n\n\n

    \n
      \n
    • \u56e0\u6570\u5206\u89e3\u3057\u3066\u6c42\u3081\u308b<\/li>\n\n\n\n
    • \u89e3\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u6c42\u3081\u308b<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

      \u89e3\u306e\u516c\u5f0f\u3092\u4f7f\u3048\u3070\u3069\u3093\u306a\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3067\u3082\u89e3\u3051\u308b\u306e\u3067\u8d85\u4fbf\u5229\uff01<\/span><\/p>\n\n\n\n

      \n
      \n

      ax2<\/sup>+bx+c=0 \u306e\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u3092\u5c0e\u51fa\u3057\u307e\u3059\u3002
      \n \u4fc2\u6570\u3092\u5165\u529b\u5f8c\u3001\u7b97\u51fa\u30dc\u30bf\u30f3\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n <\/div>\n

      \n
      \n