Giải bài 5 trang 22 sách bài tập toán 8 - Chân trời sáng tạo

Đề bài

Tính:

a) $x - \frac{{2x - y}}{4} + \frac{{x + 4y}}{{12}}$;

b) $\frac{y}{x} - \frac{x}{y} - \frac{{{x^2} + {y^2}}}{{xy}}$;

c) $\frac{4}{{x + 2}} - \frac{3}{{x - 2}} + \frac{{12}}{{{x^2} - 4}}$;

d) $\frac{{x + y}}{{{x^2} - xy}} - \frac{{4x}}{{{x^2} - {y^2}}} - \frac{{x - y}}{{{x^2} + xy}}$.

Lời giải chi tiết

a) $x - \frac{{2x - y}}{4} + \frac{{x + 4y}}{{12}} = \frac{{12x}}{{12}} - \frac{{3\left( {2x - y} \right)}}{{12}} + \frac{{x + 4y}}{{12}} = \frac{{12x - 6x + 3y + x + 4y}}{{12}} = \frac{{7x + 7y}}{{12}}$

b) $\frac{y}{x} - \frac{x}{y} - \frac{{{x^2} + {y^2}}}{{xy}} = \frac{{{y^2}}}{{xy}} - \frac{{{x^2}}}{{xy}} - \frac{{{x^2} + {y^2}}}{{xy}} = \frac{{{y^2} - {x^2} - {x^2} - {y^2}}}{{xy}} = \frac{{ - 2{x^2}}}{{xy}} = \frac{{ - 2x}}{y}$

c) $\frac{4}{{x + 2}} - \frac{3}{{x - 2}} + \frac{{12}}{{{x^2} - 4}} = \frac{{4\left( {x - 2} \right)}}{{\left( {x + 2} \right)\left( {x - 2} \right)}} - \frac{{3\left( {x + 2} \right)}}{{\left( {x + 2} \right)\left( {x - 2} \right)}} + \frac{{12}}{{\left( {x + 2} \right)\left( {x - 2} \right)}}$

$= \frac{{4x - 8 - 3x - 6 + 12}}{{\left( {x + 2} \right)\left( {x - 2} \right)}} = \frac{{x - 2}}{{\left( {x + 2} \right)\left( {x - 2} \right)}} = \frac{1}{{x + 2}}$

d) $\frac{{x + y}}{{{x^2} - xy}} - \frac{{4x}}{{{x^2} - {y^2}}} - \frac{{x - y}}{{{x^2} + xy}} = \frac{{{{\left( {x + y} \right)}^2}}}{{x\left( {x + y} \right)\left( {x - y} \right)}} - \frac{{4{x^2}}}{{x\left( {x + y} \right)\left( {x - y} \right)}} - \frac{{{{\left( {x - y} \right)}^2}}}{{x\left( {x + y} \right)\left( {x - y} \right)}}$

$= \frac{{{x^2} + 2xy + {y^2} - 4{x^2} - {x^2} + 2xy - {y^2}}}{{x\left( {x + y} \right)\left( {x - y} \right)}} = \frac{{4xy - 4{x^2}}}{{x\left( {x + y} \right)\left( {x - y} \right)}} = \frac{{ - 4x\left( {x - y} \right)}}{{x\left( {x + y} \right)\left( {x - y} \right)}} = \frac{{ - 4}}{{x + y}}$