333 Вправа 333Алгебра 8 клас Бевз 2025 · § 7 · стор. 68

Умова

Спрости вираз.

333. Спрости вираз.

а) $(1 - \frac{b^{2} - a^{2} - c^{2}}{2 a c}) : (\frac{a + c}{b} - 1) - \frac{b^{2}}{2 a c}$ ;

1) $1 - \frac{b^{2} - a^{2} - c^{2}}{2 a c} =$ $\frac{2 a c - b^{2} + a^{2} + c^{2}}{2 a c}$ =

= $\frac{{(a + c)}^{2} - b^{2}}{2 a c} =$ $\frac{(a + c - b) (a + c + b)}{2 a c}$ ;

2) $\frac{a + c}{b} - 1 =$ $\frac{a + c - b}{b}$ ;

3) $\frac{(a + c - b) (a + c + b)}{2 a c} : \frac{a + c - b}{b}$ =

= $\frac{(a + c - b) (a + c + b)}{2 a c} \cdot \frac{b}{a + c - b}$ =

= $\frac{(a + c - b) (a + c + b) \cdot b}{2 a c (a + c - b)} =$ $\frac{b (a + b + c)}{2 a c}$ ;

4) $\frac{b (a + b + c)}{2 a c} - \frac{b^{2}}{2 a c} =$ $\frac{a b + b^{2} + b c - b^{2}}{2 a c}$ =

= $\frac{a b + b c}{2 a c} =$ $\frac{b (a + c)}{2 a c}$ ;

б) $\frac{3 x^{2}}{a^{2} - x^{2}} - 3 a (\frac{1}{a - x} - \frac{x}{a^{3} - x^{3}} \cdot \frac{a^{2} + a x + x^{2}}{a + x})$ ;

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

= $- \frac{x (x^{2} + x a + a^{2})}{(x + a) (x^{2} + x a + a^{2}) (x - a)} =$ $\frac{x}{(x + a) (a - x)}$ ;

2) $\frac{1}{a - x} - \frac{x}{(x + a) (a - x)} =$ $\frac{1^{∖ a + x}}{a - x} - \frac{x}{(a - x) (a + x)}$ =

= $\frac{a + x - x}{(a - x) (a + x)} =$ $\frac{a}{(a - x) (a + x)} =$ $\frac{a}{a^{2} - x^{2}}$ ;

3) $3 a \cdot \frac{a}{(x + a) (a - x)} =$ $\frac{3 a \cdot a}{(x + a) (a - x)} =$ $\frac{3 a^{2}}{(a - x) (a + x)}$ ;

4) $\frac{3 x^{2}}{a^{2} - x^{2}} - \frac{3 a^{2}}{(x + a) (a - x)}$ =

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

= $\frac{- 3 (x - a) (x + a)}{(x - a) (x + a)} =$ $- 3$ .