Страница 116 номер 404, ГДЗ по алгебре за 9 класс к учебнику Мерзляка
1) \(x^{2} + 6x - 7 < 0\)
\(a > 0\) ветви вверх
\(\dfrac{D}{4} = 9 + 7 = 16 > 0\)
\(x_{1} = - 3 - 4 = - 7;\ x_{2} = - 3 + 4 = 1\)
\(x \in ( - 7;\ 1)\)
2) \(x^{2} - 2x - 48 \geqslant 0\)
\(a > 0\) ветви вверх
\(\dfrac{D}{4} = 1 + 48 = 49 > 0\)
\(x_{1} = 1 - 7 = - 6;\ x_{2} = 1 + 7 = 8\)
\(x \in ( - \infty; - 6] \cup [ 8; + \infty)\)
3) \(-x^{2} - 6x - 5 > 0\)
\(a < 0\) ветви вниз
\(\dfrac{D}{4} = 9 - 5 = 4 > 0\)
\(x_{1} = \dfrac{3 - 2}{- 1} = - 1;\ x_{2} = \dfrac{3 + 2}{- 1} = - 5\)
\(x \in ( - 5; - 1)\)
4) \(-x^{2} + 4x - 3 < 0\)
\(a < 0\) ветви вниз
\(\dfrac{D}{4} = 4 - 3 = 1 > 0\)
\(x_{1} = \dfrac{- 2 - 1}{- 1} = 3;\ x_{2} = \dfrac{- 2 + 1}{- 1} = 1\)
\(x \in ( - \infty;\ 1) \cup (3; + \infty)\)
5) \(3x^{2} - 7x + 4 \leqslant 0\)
\(a > 0\) ветви вверх
\(D = 49 - 4 \cdot 3 \cdot 4 = 49 - 48 = 1 > 0\)
\(x_{1} = \dfrac{7 - 1}{6} = 1;\ x_{2} = \dfrac{7 + 1}{6} = \dfrac{8}{6} = \dfrac{4}{3}\)
\(x \in \left[ 1;\dfrac{4}{3} \right]\)
6) \(2x^{2} + 3x + 1 > 0\)
\(a > 0\) ветви вверх
\(D = 9 - 4 \cdot 2 = 1\)
\(x_{1} = \dfrac{- 3 - 1}{4} = - 1;\ x_{2} = \dfrac{- 3 + 1}{4} = - \dfrac{1}{2}\)
\(x \in ( - \infty; - 1) \cup ( - 0{,}5; + \infty)\)
7) \(4x^{2} - 12x \leqslant 0\)
\(a > 0\) ветви вверх
\(4x(x - 3) \leqslant 0 \Rightarrow x_{1} = 0;\ x_{2} = 3\)
\(x \in [ 0;\ 3]\)
8) \(4x^{2} - 9 > 0\)
\(a > 0\) ветви вверх
\(D = 0 + 4 \cdot 4 \cdot 9 = 4^{2} \cdot 3^{2} = 12^{2}\)
\(x_{1} = \dfrac{- 12}{8} = - \dfrac{3}{2};\ x_{2} = \dfrac{12}{8} = \dfrac{3}{2}\)
\(x \in \left( - \infty; - \dfrac{3}{2} \right) \cup \left( \dfrac{3}{2}; + \infty \right)\)
9) \(x^{2} - 12x + 36 > 0\)
\(a > 0\) ветви вверх
\(\dfrac{D}{4} = 36 - 36 = 0\)
\(x = 6\)
\(x \in ( - \infty;\ 6) \cup (6; + \infty)\)
10) \(4x^{2} - 12x + 9 \geqslant 0\)
\(a > 0\) ветви вверх
\(\dfrac{D}{4} = 36 - 4 \cdot 9 = 0\)
\(x = \dfrac{6}{4} = \dfrac{3}{2}\)
\(x \in ( - \infty; + \infty)\)
11) \(x^{2} + 4x + 4 < 0\)
\(a > 0\) ветви вверх
\((x + 2)^{2} < 0 \Rightarrow D = 0 \Rightarrow x = - 2\)
\(x \in \varnothing\)
12) \(49x^{2} - 14x + 1 \leqslant 0\)
\(a > 0\) ветви вверх
\((7x - 1)^{2} \leqslant 0 \Rightarrow D = 0 \Rightarrow x = \dfrac{1}{7}\)
\(x = \dfrac{1}{7}\)
13) \(2x^{2} - x + 3 > 0\)
\(a > 0\) ветви вверх
\(D = 1 - 4 \cdot 2 \cdot 3 < 0\) – нет корней
\(x \in ( - \infty; + \infty)\)
14) \(3x^{2} - 4x + 5 \leqslant 0\)
\(a > 0\) ветви вверх
\(\dfrac{D}{4} = 4 - 3 \cdot 5 < 0\)
\(x \in \varnothing\)
15) \(- 4x^{2} + 5x - 7 > 0\)
\(a < 0\) ветви вниз
\(D = 25 + 4 \cdot ( - 4) \cdot 7 < 0\) нет корней
\(x \in \varnothing\)
16) \(- 2x^{2} + 3x - 2 \leqslant 0\)
\(a < 0\) ветви вниз
\(D = 9 + 4 \cdot ( - 2) \cdot 2 = 9 - 16 < 0\)
\(x \in ( - \infty; + \infty)\)
Ответ: 1) \(x \in ( - 7;\ 1)\); 2) \(x \in ( - \infty; - 6] \cup [ 8; + \infty)\); 3) \(x \in ( - 5; - 1)\); 4) \(x \in ( - \infty;\ 1) \cup (3; + \infty)\); 5) \(x \in \left[ 1;\frac{4}{3} \right]\); 6) \(x \in ( - \infty; - 1) \cup ( - 0{,}5; + \infty)\); 7) \(x \in [ 0;\ 3]\); 8) \(x \in \left( - \infty; - \frac{3}{2} \right) \cup \left( \frac{3}{2}; + \infty \right)\); 9) \(x \in ( - \infty;\ 6) \cup (6; + \infty)\); 10) \(x \in ( - \infty; + \infty)\); 11) \(x \in \varnothing\); 12) \(x = \frac{1}{7}\); 13) \(x \in ( - \infty; + \infty)\); 14) \(x \in \varnothing\); 15) \(x \in \varnothing\); 16) \(x \in ( - \infty; + \infty)\).
