こんにちは。今回は2次関数と2次方程式の解ということで, 2次方程式の解の範囲をグラフ的に捉えて解決していきましょう。最後に数IIでの解法も載せておきます。
【例題】2次方程式が, 次のような解をもつとき, 定数
の範囲を求めよ。
(ア) 異なる2つの解がともに1より大きいとき
(イ) 異なる2つの解がともに1より小さいとき
(ウ) 1つの解が1より大きく, 他の解が1より小さいとき
とおく。
の関数
のグラフと
軸との交点が2次方程式の解になることを利用して解いていく。このとき, 関数
のイメージとしては以下のようになればよい。
解決方法は次の3つを調べること。
それは, 判別式, 軸,
![Rendered by QuickLaTeX.com f(1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fb9bb7d8ecc863e35618710125983a3f_l3.png)
判別式
![Rendered by QuickLaTeX.com (D)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8dae430f66b59ef4eabb4fe466018144_l3.png)
軸
![Rendered by QuickLaTeX.com (\ell)>1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-08b1878ffae03fdf403c081ec605d70d_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
![Rendered by QuickLaTeX.com f(1)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2a1bdcc792f20576b3a1ea6489060094_l3.png)
![Rendered by QuickLaTeX.com D>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8e789aabbd3606a416f1a8f3d0acd7b5_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
この3つを同時に満たすことで,
![Rendered by QuickLaTeX.com f(x)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d2fed79cedb38d253149db96430fffcb_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
まず, 判別式
![Rendered by QuickLaTeX.com D/4=(2m)^2-2(m+3)=4m^2-2m-6>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2c5326099c4c939a69fb143562899a1a_l3.png)
![Rendered by QuickLaTeX.com (m+1)(2m-3)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9914c21c8419fc70030b16b10eeaa92b_l3.png)
![Rendered by QuickLaTeX.com m<-1, m>\dfrac32\cdots\maru1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-749d7a7a87dfbd4d7b1b1416473efca6_l3.png)
次に軸に関して,
![Rendered by QuickLaTeX.com f(x)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d2fed79cedb38d253149db96430fffcb_l3.png)
![Rendered by QuickLaTeX.com f(x)=2(x-m)^2-2m^2+m+3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-824c2dcc597b8d59f259d929af2319e0_l3.png)
軸の式は
![Rendered by QuickLaTeX.com x=m](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-71310bc6b9c3b57a7e74266e5907807f_l3.png)
これが1より大きいことが条件なので,
![Rendered by QuickLaTeX.com m>1\cdots\maru2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5cf4aec53e734f3333a2ad0d9a193e28_l3.png)
最後に,
![Rendered by QuickLaTeX.com f(1)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2a1bdcc792f20576b3a1ea6489060094_l3.png)
![Rendered by QuickLaTeX.com f(1)=-3m+5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1e8d89204b73dc39d4869a45c5f188c9_l3.png)
![Rendered by QuickLaTeX.com -3m+5>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f95574c5718035e4c8c80002dbb604c0_l3.png)
![Rendered by QuickLaTeX.com m<\dfrac53\cdots\maru3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a11c78a12597613404f205676eea3416_l3.png)
![Rendered by QuickLaTeX.com \maru1, \maru2, \maru3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a94935c102c23549ddb1822473356c2d_l3.png)
![Rendered by QuickLaTeX.com \dfrac32<m<\dfrac53\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-27987518501fe91578ec41df6f29970b_l3.png)
(イ)の解法
(ア)の解法同様, グラフを描いてイメージをつかむ。
このとき, 関数のイメージとしては以下のようになればよい。
解決方法は次の3つを調べること。
それは, 判別式, 軸,
![Rendered by QuickLaTeX.com f(1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fb9bb7d8ecc863e35618710125983a3f_l3.png)
判別式
![Rendered by QuickLaTeX.com (D)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8dae430f66b59ef4eabb4fe466018144_l3.png)
軸
![Rendered by QuickLaTeX.com (\ell)<1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3b654906c5cd60aace22e9baade36d01_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
![Rendered by QuickLaTeX.com f(1)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2a1bdcc792f20576b3a1ea6489060094_l3.png)
![Rendered by QuickLaTeX.com D>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8e789aabbd3606a416f1a8f3d0acd7b5_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
この3つを同時に満たすことで,
![Rendered by QuickLaTeX.com f(x)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d2fed79cedb38d253149db96430fffcb_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
まず, 判別式
![Rendered by QuickLaTeX.com D/4=(2m)^2-2(m+3)=4m^2-2m-6>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2c5326099c4c939a69fb143562899a1a_l3.png)
![Rendered by QuickLaTeX.com (m+1)(2m-3)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9914c21c8419fc70030b16b10eeaa92b_l3.png)
![Rendered by QuickLaTeX.com m<-1, m>\dfrac32\cdots\maru1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-749d7a7a87dfbd4d7b1b1416473efca6_l3.png)
次に軸に関して,
![Rendered by QuickLaTeX.com f(x)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d2fed79cedb38d253149db96430fffcb_l3.png)
![Rendered by QuickLaTeX.com f(x)=2(x-m)^2-2m^2+m+3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-824c2dcc597b8d59f259d929af2319e0_l3.png)
軸の式は
![Rendered by QuickLaTeX.com x=m](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-71310bc6b9c3b57a7e74266e5907807f_l3.png)
これが1より小さいことが条件なので,
![Rendered by QuickLaTeX.com m<1\cdots\maru2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-42d5d8ac7a54422162a17d1ed300617d_l3.png)
最後に,
![Rendered by QuickLaTeX.com f(1)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2a1bdcc792f20576b3a1ea6489060094_l3.png)
![Rendered by QuickLaTeX.com f(1)=-3m+5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1e8d89204b73dc39d4869a45c5f188c9_l3.png)
![Rendered by QuickLaTeX.com -3m+5>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f95574c5718035e4c8c80002dbb604c0_l3.png)
![Rendered by QuickLaTeX.com m<\dfrac53\cdots\maru3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a11c78a12597613404f205676eea3416_l3.png)
![Rendered by QuickLaTeX.com \maru1, \maru2, \maru3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a94935c102c23549ddb1822473356c2d_l3.png)
![Rendered by QuickLaTeX.com m<-1\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2e3a81e851cedb089bd8b0e73df0f590_l3.png)
(ウ)の解法
これまでと同様にグラフを描いてイメージをつかむ。
上の図からわかるように, 1つの解は1より大きく, 他の解は1より小さい場合では,
![Rendered by QuickLaTeX.com f(1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fb9bb7d8ecc863e35618710125983a3f_l3.png)
![Rendered by QuickLaTeX.com f(1)<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-0ffecb96590a4239f9e00b2c2b4f4415_l3.png)
![Rendered by QuickLaTeX.com f(1)=-3m+5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1e8d89204b73dc39d4869a45c5f188c9_l3.png)
![Rendered by QuickLaTeX.com -3m+5<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-463c425729f14249d39597f2d1e7749a_l3.png)
![Rendered by QuickLaTeX.com m>\dfrac53](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-48ebf251a7b0ccaadeb46d9802fe5bb3_l3.png)
![Rendered by QuickLaTeX.com m>\dfrac53\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c146aa9a0ab7e6ecbbd26619e97565e1_l3.png)
考え方は2次関数と2次方程式①と同じで, この①では0より大きい異なる2解だったのが今回は1より大きい異なる2解に変わっただけである。0より小さい異なる2解も同じこと。正と負の解については, 0より大きい解と0より小さい解と言い換えれば, 今回のものは1より大きい解と1より小さい解となっただけである。
![](https://mathtext.info/blog/wp-content/uploads/2022/04/snaga2jihoteikaiun161-160x92.png)
![](https://mathtext.info/blog/wp-content/uploads/2022/04/mojihakainnitei21siki-160x92.png)