簡約 (a-1)/(a-2)-(a-2)/(a+3)+1/(a-1)

解

簡約

\frac{a - 1}{a - 2} - \frac{a - 2}{a + 3} + \frac{1}{a - 1}

\frac{7 a ^{2} - 12 a + 1}{( a - 2 ) ( a + 3 ) ( a - 1 )}

解答ステップ

\frac{a - 1}{a - 2} - \frac{a - 2}{a + 3} + \frac{1}{a - 1}

以下の最小公倍数:

a - 2, a + 3, a - 1 : (a - 2) (a + 3) (a - 1)

LCMに基づいて分数を調整する

= \frac{( a + 3 ) ( a - 1 ) ^{2}}{( a - 2 ) ( a + 3 ) ( a - 1 )} - \frac{( a - 2 ) ^{2} ( a - 1 )}{( a - 2 ) ( a + 3 ) ( a - 1 )} + \frac{( a - 2 ) ( a + 3 )}{( a - 2 ) ( a + 3 ) ( a - 1 )}

分数の規則を適用する:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{( a + 3 ) ( a - 1 ) ^{2} - ( a - 2 ) ^{2} ( a - 1 ) + ( a - 2 ) ( a + 3 )}{( a - 2 ) ( a + 3 ) ( a - 1 )}

簡素化

(a + 3) (a - 1)^{2} - (a - 2)^{2} (a - 1) + (a - 2) (a + 3) : 7 a^{2} - 12 a + 1

= \frac{7 a ^{2} - 12 a + 1}{( a - 2 ) ( a + 3 ) ( a - 1 )}