540. Решите уравнение: в) $$5x^{2}-9x + 2$$; 6) $$-t^{2}=5t-14$$; в) $$6x+9=x^{2}$$; r) $$x-5=x^{2}-25$$; е) $$15y^{2}-30=-7y$$; ж) $$25p^{2}=10p-1$$; з) $$299x^{2}+100x=500-101x^{2}$$.

Решим уравнения:

в) $$5x^2 - 9x + 2 = 0$$

$$D = (-9)^2 - 4 \cdot 5 \cdot 2 = 81 - 40 = 41$$

$$x_1 = \frac{9 + \sqrt{41}}{10}, \quad x_2 = \frac{9 - \sqrt{41}}{10}$$

Ответ: $$x_1 = \frac{9 + \sqrt{41}}{10}, \quad x_2 = \frac{9 - \sqrt{41}}{10}$$

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б) $$-t^2 = 5t - 14$$

$$t^2 + 5t - 14 = 0$$

$$D = 5^2 - 4 \cdot 1 \cdot (-14) = 25 + 56 = 81$$

$$t_1 = \frac{-5 + \sqrt{81}}{2} = \frac{-5 + 9}{2} = \frac{4}{2} = 2$$

$$t_2 = \frac{-5 - \sqrt{81}}{2} = \frac{-5 - 9}{2} = \frac{-14}{2} = -7$$

Ответ: $$t_1 = 2, t_2 = -7$$

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в) $$6x + 9 = x^2$$

$$x^2 - 6x - 9 = 0$$

$$D = (-6)^2 - 4 \cdot 1 \cdot (-9) = 36 + 36 = 72$$

$$x_1 = \frac{6 + \sqrt{72}}{2} = \frac{6 + 6\sqrt{2}}{2} = 3 + 3\sqrt{2}$$

$$x_2 = \frac{6 - \sqrt{72}}{2} = \frac{6 - 6\sqrt{2}}{2} = 3 - 3\sqrt{2}$$

Ответ: $$x_1 = 3 + 3\sqrt{2}, x_2 = 3 - 3\sqrt{2}$$

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г) $$x - 5 = x^2 - 25$$

$$x^2 - x - 20 = 0$$

$$D = (-1)^2 - 4 \cdot 1 \cdot (-20) = 1 + 80 = 81$$

$$x_1 = \frac{1 + \sqrt{81}}{2} = \frac{1 + 9}{2} = \frac{10}{2} = 5$$

$$x_2 = \frac{1 - \sqrt{81}}{2} = \frac{1 - 9}{2} = \frac{-8}{2} = -4$$

Ответ: $$x_1 = 5, x_2 = -4$$

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д) $$15y^2 - 30 = -7y$$

$$15y^2 + 7y - 30 = 0$$

$$D = 7^2 - 4 \cdot 15 \cdot (-30) = 49 + 1800 = 1849$$

$$y_1 = \frac{-7 + \sqrt{1849}}{30} = \frac{-7 + 43}{30} = \frac{36}{30} = \frac{6}{5} = 1.2$$

$$y_2 = \frac{-7 - \sqrt{1849}}{30} = \frac{-7 - 43}{30} = \frac{-50}{30} = -\frac{5}{3} = -1\frac{2}{3}$$

Ответ: $$y_1 = 1.2, y_2 = -1\frac{2}{3}$$

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ж) $$25p^2 = 10p - 1$$

$$25p^2 - 10p + 1 = 0$$

$$D = (-10)^2 - 4 \cdot 25 \cdot 1 = 100 - 100 = 0$$

$$p = \frac{10 \pm \sqrt{0}}{50} = \frac{10}{50} = \frac{1}{5} = 0.2$$

Ответ: $$p = 0.2$$

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з) $$299x^2 + 100x = 500 - 101x^2$$

$$299x^2 + 101x^2 + 100x - 500 = 0$$

$$400x^2 + 100x - 500 = 0$$

$$4x^2 + x - 5 = 0$$

$$D = 1^2 - 4 \cdot 4 \cdot (-5) = 1 + 80 = 81$$

$$x_1 = \frac{-1 + \sqrt{81}}{8} = \frac{-1 + 9}{8} = \frac{8}{8} = 1$$

$$x_2 = \frac{-1 - \sqrt{81}}{8} = \frac{-1 - 9}{8} = \frac{-10}{8} = -\frac{5}{4} = -1.25$$

Ответ: $$x_1 = 1, x_2 = -1.25$$