Упражнение 95

\({\largeа)}\ \frac{b-6}{4-b^2}+\frac{2}{2b-b^2}=\frac{b-6}{ (2-b)(2+b)}+\frac{2}{ b(2-b)}=\frac{ (b-6)\cdot{b}+2\cdot(2+b)}{ b(2-b)(2+b)}=\frac{b^2-6b+4+2b}{ b(2-b)(2+b)}=\frac{4-4b+b^2}{ b(2-b)(2+b)}=\frac{ (2-b)^2}{ b(2-b)(2+b)}=\frac{2-b}{ b(2+b)};\)

\({\largeб)}\ \frac{b}{ab-5a^2}-\frac{15b-25a}{b^2-25a^2}=\frac{b}{ a(b-5a)}-\frac{15b-25a}{ (b-5a)(b+5a)}=\frac{ b\cdot(b+5a)-(15b-25a)\cdot{a}}{ a(b-5a)(b+5a)}=\frac{b^2+5ab-15ab+25a^2}{ a(b-5a)(b+5a)}=\frac{b^2-10ab+25a^2}{ a(b-5a)(b+5a)}=\frac{ (b-5a)^2}{ a(b-5a)(b+5a)}=\frac{b-5a}{ a(b+5a)};\)

\({\largeв)}\ \frac{x-12a}{x^2-16a^2}-\frac{4a}{4ax-x^2}=\frac{x-12a}{ (x-4a)(x+4a)}-\frac{4a}{ x(4a-x)}=\frac{x-12a}{ (x-4a)(x+4a)}+\frac{4a}{ x(x-4a)}=\frac{ (x-12a)\cdot{x}+4a\cdot(x+4a)}{ x(x-4a)(x+4a)}=\frac{x^2-12ax+4ax+16a^2}{ x(x-4a)(x+4a)}=\frac{x^2-8ax+16a^2}{ x(x-4a)(x+4a)}=\frac{ (x-4a)^2}{ x(x-4a)(x+4a)}=\frac{x-4a}{ x(x+4a)};\)

\({\largeг)}\ \frac{a-30y}{a^2-100y^2}-\frac{10y}{10ay-a^2}=\frac{a-30y}{ (a-10y)(a+10y)}-\frac{10y}{ a(10y-a)}=\frac{a-30y}{ (a-10y)(a+10y)}+\frac{10y}{ a(a-10y)}=\frac{ (a-30y)\cdot{a}+10y\cdot(a+10y)}{ a(a-10y)(a+10y)}=\frac{a^2-30ay+10ay+100y^2}{ a(a-10y)(a+10y)}=\frac{a^2-20ay+100y^2}{ a(a-10y)(a+10y)}=\frac{ (a-10y)^2}{ a(a-10y)(a+10y)}=\frac{a-10y}{ a(a+10y)}.\)