ВПР 1 №33 Алгебра = ВПТ 1 №33 Математика
Спростіть вираз:
1. $\frac{4m+18}{m^2-9}-\frac{5}{m-3}-\frac{1}{m+3};$
2. $\frac{2x}{2x+3}+\frac{5}{3-2x}-\frac{4x^2+9}{4x^2-9};$
3. $\frac{9x}{3x+2y^2}-\frac{4y}{3x^2+2xy};$
4. $\frac{4a}{4a^2-1}-\frac{2a+1}{6a-3}+\frac{2a-1}{4a+2};$
5. $\frac{2x-1}{x^2+x+1}+\frac{4x^2+3x-7}{x^3-1};$
6. $\frac{a^2}{3ab-2-a+6b}-\frac{a}{3b-1}.$
Розв'язок:
1. $\frac{4m+18}{m^2-9}-\frac{5}{m-3}-\frac{1}{m+3}=$
$= \frac{4m+18}{\left(m-3\right)\left(m+3\right)}-\frac{5}{m-3}+\frac{1}{m+3}=$
$= \frac{\left(4m+18\right)-5\left(m+3\right)-\left(m-3\right)}{\left(m-3\right)\left(m+3\right)}=$
$= \frac{4m+18-5m-15+m-3}{\left(m-3\right)\left(m+3\right)}=$
$= \frac{0}{m^2-9}=0;$
2. $\frac{2x}{2x+3}+\frac{5}{3-2x}-\frac{4x^2+9}{4x^2-9}=$
$= \frac{2x}{2x+3}-\frac{5}{2-3x}-$
$- \frac{4x^2+9}{\left(2x-3\right)\left(2x+3\right)}=$
$= \frac{2x\left(2x-3\right)-5\left(2x+3\right)-\left(4x^2+9\right)}{\left(2x-3\right)\left(2x+3\right)}=$
$= \frac{4x^2-6x-10x-15-4x^2-9}{\left(2x-3\right)\left(2x+3\right)}=$
$= \frac{-16x-24}{\left(2x-3\right)\left(2x+3\right)}= \frac{-8\left(2x+3\right)}{\left(2x-3\right)\left(2x+3\right)}=$
$= -\frac{8}{2x-3}=\frac{8}{3-2x};$
3. $\frac{9x}{3x+2y^2}-\frac{4y}{3x^2+2xy}=$
$= \frac{9x}{y\left(3x+2y\right)}-\frac{4y}{x\left(3x+2y\right)}=$
$= \frac{9x}{y\left(3x+2y\right)}=\frac{9x^2-4y^2}{xy\left(3x+2y\right)}=$
$= \frac{\left(3x-2y\right)\left(3x+2y\right)}{xy\left(3x+2y\right)}=\frac{3x-2y}{xy}.$
4. $\frac{4a}{4a^2-1}-\frac{2a+1}{6a-3}+\frac{2a-1}{4a+2}=$
$= \frac{4a}{\left(2a-1\right)\left(2a+1\right)}-\frac{2a+1}{3\left(2a-1\right)}+$
$+ \frac{2a-1}{2\left(2a+1\right)}=$
$=\frac{24a-2\left(2a+1\right)^2+3\left(2a-1\right)^2}{6\left(2a-1\right)\left(2a+1\right)}=$
$= \frac{24a-2\left(4a^2+4a+1\right)+3\left(4a^2-4a+1\right)}{6\left(2a-1\right)\left(2a+1\right)}=$
$= \frac{24a-8a^2-8a-2+12a^2-12a+3}{6\left(2a-1\right)\left(2a+1\right)}=$
$= \frac{4a^2+4a+1}{6\left(2a-1\right)\left(2a+1\right)}=$
$= \frac{\left(2a+1\right)^2}{6\left(2a-1\right)\left(2a+1\right)}= \frac{2a+1}{6\left(2a-1\right)}.$
5. $\frac{2x-1}{x^2+x+1}+\frac{4x^2+3x-7}{x^3-1}=$
$= \frac{2x-1}{x^2+x+1}+\frac{4x^2+3x-7}{\left(x-1\right)\left(x^2+x+1\right)}=$
$= \frac{\left(x-1\right)\left(2x-1\right)+\left(4x^2+3x-7\right)}{\left(x-1\right)\left(x^2+x+1\right)}=$
$= \frac{2x^2-x-2x+1+4x^2+3x-7}{\left(x-1\right)\left(x^2+x+1\right)}=$
$= \frac{6x^2-6}{\left(x-1\right)\left(x^2+x+1\right)}=$
$= \frac{6\left(x^2-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=$
$= \frac{6\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}= \frac{6\left(x+1\right)}{x^2+x+1}.$
6. $\frac{a^2}{3ab-2-a+6b}-\frac{a}{3b-1}=$
$= \frac{a^2}{\left(3ab-a\right)-\left(2-6b\right)}-\frac{a}{3b-1}=$
$= \frac{a^2}{a\left(3b-1\right)-2\left(1-3b\right)}-\frac{a}{3b-1}=$
$= \frac{a^2}{a\left(3b-1\right)+2\left(3b-1\right)}-\frac{a}{3b-1}=$
$= \frac{a^2}{\left(3b-1\right)\left(a+2\right)}-\frac{a}{3b-1}=$
$= \frac{a^2-a\left(a+2\right)}{\left(3b-1\right)\left(a+2\right)}= \frac{a^2-a^2-2a}{\left(3b-1\right)\left(a+2\right)}=$
$= -\frac{2a}{\left(3b-1\right)\left(a+2\right)}= \frac{2a}{\left(1-3b\right)\left(a+2\right)}.$