542. Решите уравнение: a) 5x2 = 9x + 2; 6) -x² = 5x-14; в) 6x + 9 = x²; г) 2 - 5 = z² - 25; д) у² = 52у - 576; e) 15y2 30 = 22y + 7; ж) 25р² = 10р 1; 3) 299x² + 100x = 500 - 101x².

Решение уравнений:

1. a) $$5x^2 = 9x + 2$$
   $$5x^2 - 9x - 2 = 0$$
   $$D = (-9)^2 - 4 \cdot 5 \cdot (-2) = 81 + 40 = 121$$
   $$x_1 = \frac{9 + \sqrt{121}}{2 \cdot 5} = \frac{9 + 11}{10} = \frac{20}{10} = 2$$
   $$x_2 = \frac{9 - \sqrt{121}}{2 \cdot 5} = \frac{9 - 11}{10} = \frac{-2}{10} = -0.2$$
   Ответ: $$x_1 = 2$$, $$x_2 = -0.2$$.
2. б) $$-x^2 = 5x - 14$$
   $$x^2 + 5x - 14 = 0$$
   $$D = 5^2 - 4 \cdot 1 \cdot (-14) = 25 + 56 = 81$$
   $$x_1 = \frac{-5 + \sqrt{81}}{2 \cdot 1} = \frac{-5 + 9}{2} = \frac{4}{2} = 2$$
   $$x_2 = \frac{-5 - \sqrt{81}}{2 \cdot 1} = \frac{-5 - 9}{2} = \frac{-14}{2} = -7$$
   Ответ: $$x_1 = 2$$, $$x_2 = -7$$.
3. в) $$6x + 9 = x^2$$
   $$x^2 - 6x - 9 = 0$$
   $$D = (-6)^2 - 4 \cdot 1 \cdot (-9) = 36 + 36 = 72$$
   $$x_1 = \frac{6 + \sqrt{72}}{2 \cdot 1} = \frac{6 + 6\sqrt{2}}{2} = 3 + 3\sqrt{2}$$
   $$x_2 = \frac{6 - \sqrt{72}}{2 \cdot 1} = \frac{6 - 6\sqrt{2}}{2} = 3 - 3\sqrt{2}$$
   Ответ: $$x_1 = 3 + 3\sqrt{2}$$, $$x_2 = 3 - 3\sqrt{2}$$.
4. г) $$z - 5 = z^2 - 25$$
   $$z^2 - z - 20 = 0$$
   $$D = (-1)^2 - 4 \cdot 1 \cdot (-20) = 1 + 80 = 81$$
   $$z_1 = \frac{1 + \sqrt{81}}{2 \cdot 1} = \frac{1 + 9}{2} = \frac{10}{2} = 5$$
   $$z_2 = \frac{1 - \sqrt{81}}{2 \cdot 1} = \frac{1 - 9}{2} = \frac{-8}{2} = -4$$
   Ответ: $$z_1 = 5$$, $$z_2 = -4$$.
5. д) $$y^2 = 52y - 576$$
   $$y^2 - 52y + 576 = 0$$
   $$D = (-52)^2 - 4 \cdot 1 \cdot 576 = 2704 - 2304 = 400$$
   $$y_1 = \frac{52 + \sqrt{400}}{2 \cdot 1} = \frac{52 + 20}{2} = \frac{72}{2} = 36$$
   $$y_2 = \frac{52 - \sqrt{400}}{2 \cdot 1} = \frac{52 - 20}{2} = \frac{32}{2} = 16$$
   Ответ: $$y_1 = 36$$, $$y_2 = 16$$.
6. e) $$15y^2 - 30 = 22y + 7$$
   $$15y^2 - 22y - 37 = 0$$
   $$D = (-22)^2 - 4 \cdot 15 \cdot (-37) = 484 + 2220 = 2704$$
   $$y_1 = \frac{22 + \sqrt{2704}}{2 \cdot 15} = \frac{22 + 52}{30} = \frac{74}{30} = \frac{37}{15}$$
   $$y_2 = \frac{22 - \sqrt{2704}}{2 \cdot 15} = \frac{22 - 52}{30} = \frac{-30}{30} = -1$$
   Ответ: $$y_1 = \frac{37}{15}$$, $$y_2 = -1$$.
7. ж) $$25p^2 = 10p - 1$$
   $$25p^2 - 10p + 1 = 0$$
   $$D = (-10)^2 - 4 \cdot 25 \cdot 1 = 100 - 100 = 0$$
   $$p = \frac{10}{2 \cdot 25} = \frac{10}{50} = \frac{1}{5} = 0.2$$
   Ответ: $$p = 0.2$$.
8. з) $$299x^2 + 100x = 500 - 101x^2$$
   $$400x^2 + 100x - 500 = 0$$
   $$4x^2 + x - 5 = 0$$
   $$D = 1^2 - 4 \cdot 4 \cdot (-5) = 1 + 80 = 81$$
   $$x_1 = \frac{-1 + \sqrt{81}}{2 \cdot 4} = \frac{-1 + 9}{8} = \frac{8}{8} = 1$$
   $$x_2 = \frac{-1 - \sqrt{81}}{2 \cdot 4} = \frac{-1 - 9}{8} = \frac{-10}{8} = -1.25$$
   Ответ: $$x_1 = 1$$, $$x_2 = -1.25$$.