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つの正の解
(イ) 異なる2つの負の解
(ウ) 1つは正の解で, 他の解は負の解
(ア)の解法とおく。
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(0)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-707c3265e46299a5c53620645c261a9e_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)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1d595decb7d5a632d962868061f62887_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com f(0)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dc698525d0926a3994de62d41d00b3e6_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 y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_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=(3m-1)^2-(9m^2-4)=-6m+5>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-de458fb5631e42de05fb2151422d72ac_l3.png)
![Rendered by QuickLaTeX.com m<\dfrac56\cdots\maru1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b98826b91ce2257fc47dc47dfb210592_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)=(x+3m-1)^2+6m-5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6b5005adda213dc037f89d916b9e325b_l3.png)
軸の式は
![Rendered by QuickLaTeX.com x=-3m+1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dc84ff190a8d645e966db95f76d00ffe_l3.png)
これが正なので,
![Rendered by QuickLaTeX.com -3m+1>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-30a880bb7322dab6f8b0868fcd13623f_l3.png)
![Rendered by QuickLaTeX.com m<\dfrac13\cdots\maru2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6cf25f952a8ebd0dc45506eeb1c0cb11_l3.png)
最後に,
![Rendered by QuickLaTeX.com f(0)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dc698525d0926a3994de62d41d00b3e6_l3.png)
![Rendered by QuickLaTeX.com f(0)=9m^2-4](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9f1e928526ef6e6dd7ec1c1fa662f58b_l3.png)
![Rendered by QuickLaTeX.com 9m^2-4>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c270b570312d9fede6935bb1a781d7ae_l3.png)
![Rendered by QuickLaTeX.com (3m+2)(3m-2)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1e95d14e7ec7bbb0250dab34a204f738_l3.png)
![Rendered by QuickLaTeX.com m<-\dfrac23, m>\dfrac23\cdots\maru3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c31ed4b45d1b319324baab181b1a4bb1_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<-\dfrac23\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dbac4b757360b300a48ebb5ef5560e87_l3.png)
(イ)の解法
(ア)同様にグラフを描いてイメージをつかむ。
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(0)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-707c3265e46299a5c53620645c261a9e_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)<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9ddf9f3ad47aa08caaf69168907cdeae_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com f(0)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dc698525d0926a3994de62d41d00b3e6_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 y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_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=(3m-1)^2-(9m^2-4)=-6m+5>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-de458fb5631e42de05fb2151422d72ac_l3.png)
![Rendered by QuickLaTeX.com m<\dfrac56\cdots\maru1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b98826b91ce2257fc47dc47dfb210592_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)=(x+3m-1)^2+6m-5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6b5005adda213dc037f89d916b9e325b_l3.png)
軸の式は
![Rendered by QuickLaTeX.com x=-3m+1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dc84ff190a8d645e966db95f76d00ffe_l3.png)
これが負なので,
![Rendered by QuickLaTeX.com -3m+1<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e065f8d369f7ea927fa4a113e99b2618_l3.png)
![Rendered by QuickLaTeX.com m>\dfrac13\cdots\maru2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2175d396c4f204b7e3d10aa3c311a905_l3.png)
最後に,
![Rendered by QuickLaTeX.com f(0)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dc698525d0926a3994de62d41d00b3e6_l3.png)
![Rendered by QuickLaTeX.com f(0)=9m^2-4](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9f1e928526ef6e6dd7ec1c1fa662f58b_l3.png)
![Rendered by QuickLaTeX.com 9m^2-4>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c270b570312d9fede6935bb1a781d7ae_l3.png)
![Rendered by QuickLaTeX.com (3m+2)(3m-2)>0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1e95d14e7ec7bbb0250dab34a204f738_l3.png)
![Rendered by QuickLaTeX.com m<-\dfrac23, m>\dfrac23\cdots\maru3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c31ed4b45d1b319324baab181b1a4bb1_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 \dfrac23<m<\dfrac56\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fadbfaa798a0b92996aacfa440b3dfa3_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つは正の解で, 他の解は負の解の場合,
![Rendered by QuickLaTeX.com f(0)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-707c3265e46299a5c53620645c261a9e_l3.png)
![Rendered by QuickLaTeX.com f(0)<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8e29c90dd153501d9b9804bb739d7931_l3.png)
![Rendered by QuickLaTeX.com f(0)=9m^2-4](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9f1e928526ef6e6dd7ec1c1fa662f58b_l3.png)
![Rendered by QuickLaTeX.com 9m^2-4<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8873ffd3ce45e3c68431370bbf66bf24_l3.png)
![Rendered by QuickLaTeX.com (3m+2)(3m-2)<0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c83110fedebaf33021780bacfe40b33d_l3.png)
![Rendered by QuickLaTeX.com -\dfrac23<m<\dfrac23\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-298509b69c89cbd2d3b3d9d8dd4b002a_l3.png)
流れをつかんでおこう
判別式, 軸, の値を調べて, 条件に合わせて範囲を決めていく。
![](https://mathtext.info/blog/wp-content/uploads/2022/04/mojihakainnitei211siki-160x92.png)