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≪原理≫
AB=0ならばA=0またはB=0
↑ 2つのものを掛けて0になるときは,どちらかが0だと言える ≪二次方程式に適用した場合≫
(x−a)(x−b)=0ならばx−a=0またはx−b=0
↑ 2次方程式を右辺が0の形にしておいて,左辺を因数分解すると,2つの1次方程式に分けられるところがミソ 1次方程式の解き方は中学校1年生で習い,移項や割り算だけでできる. ![]() ○1次方程式は,移項や割り算だけで解ける.
【例】
○2次方程式は,移項や割り算だけでは解けないが,因数分解したら2つの1次方程式に分けられて解ける.
x−3=0 → x=3 2x=8 → x=4
【例】
x2−3x+2=0 → ×x2=3x−2…これでは解けない x2−3x+2=0 → ○(x−1)(x−2)=0…因数分解すると → x−1=0またはx−2=0 → x=1またはx=2…解ける ![]() |
○(注意すべきポイント)
(1) 右辺=0の形に変形にすることが重要
「AB=0ならばA=0またはB=0」のように2つに分けられるのは,右辺=0の場合です.
右辺=0以外の形,例えば 「AB=2ならばA=1またはB=2」などとは言えません.
【間違い答案の例】
x2−3x+2=0 → x2−3x=−2 → x(x−3)=−2 → x=−1またはx=2×××
(2) 「左辺を因数分解する」ことが重要
因数分解とは,大雑把に言えば展開の逆だということがありますが,正確に言えば「一番大きな区切りが積(掛け算)になっている式」でなければなりません.
×次のような変形は因数分解ではありませんので,この変形で2次方程式を因数分解の方法で解くことはできません. x2+2x+4=(x+1)2+3
↑一番大きな区切りが足し算(+)になっています
x2−3x−4=x(x−3)−4
↑一番大きな区切りが引き算(−)になっています
◎次の変形は一番大きな区切りが積(掛け算)になっていて,因数分解になっています x2+5x+4=(x+1)(x+4)
↑一番大きな区切りが掛け算になっています
x2−3x=x(x−3)
↑一番大きな区切りが掛け算になっています
(3) 2つの1次方程式に分けた後に,移項すると符号が逆になることに注意
【例】
(x+3)(x+4)=0 → x+3=0またはx+4=0 → x=−3またはx=−4 (x+3)(x−4)=0 → x+3=0またはx−4=0 → x=−3またはx=4 (x−3)(x−4)=0 → x−3=0またはx−4=0 → x=3またはx=4
【要点】・・・因数分解を使って2次方程式を解く方法
(1) 右辺が0になるように変形する (2) 左辺を因数分解する(一番大きな区切りを掛け算にする) (3) 2つの1次方程式に分かれた後で,符号に注意する ※(読み飛ばしてもよい) この場面では,「x=3またはx=4」を「x=3 , 4」のように略す.この場合,カンマは「または」の意味に使っている. |
《問題》 次の二次方程式を因数分解によって解きなさい. ○はじめに,左から方程式を1つ選び,続けて右からその解を選びなさい.正しければ消えます.間違ったときは,解答欄を連打するのではなく,問題を選び直すことから始めなさい. ○間違ったときは,[HELP]ボタンが表示されます. ○HELPを読む場合でも読まない場合でも新たに問題を選択すれば,再開できます. |
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HELP![]() |
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