541. Решите уравнение: a) $$25 = 26x - x^{2}$$; б) $$3t^{2} = 10 - 29t$$; в) $$y^{2} = 4y + 96$$; г) $$3p^{2} + 3 = 10p$$; д) $$x^{2} - 20x = 20x + 100$$; е) $$25x^{2} - 13x = 10x^{2} - 7$$.

Решим уравнения:

a) $$25 = 26x - x^2$$

$$x^2 - 26x + 25 = 0$$

$$D = (-26)^2 - 4 \cdot 1 \cdot 25 = 676 - 100 = 576$$

$$x_1 = \frac{26 + \sqrt{576}}{2} = \frac{26 + 24}{2} = \frac{50}{2} = 25$$

$$x_2 = \frac{26 - \sqrt{576}}{2} = \frac{26 - 24}{2} = \frac{2}{2} = 1$$

Ответ: $$x_1 = 25, x_2 = 1$$

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б) $$3t^2 = 10 - 29t$$

$$3t^2 + 29t - 10 = 0$$

$$D = (29)^2 - 4 \cdot 3 \cdot (-10) = 841 + 120 = 961$$

$$t_1 = \frac{-29 + \sqrt{961}}{6} = \frac{-29 + 31}{6} = \frac{2}{6} = \frac{1}{3}$$

$$t_2 = \frac{-29 - \sqrt{961}}{6} = \frac{-29 - 31}{6} = \frac{-60}{6} = -10$$

Ответ: $$t_1 = \frac{1}{3}, t_2 = -10$$

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в) $$y^2 = 4y + 96$$

$$y^2 - 4y - 96 = 0$$

$$D = (-4)^2 - 4 \cdot 1 \cdot (-96) = 16 + 384 = 400$$

$$y_1 = \frac{4 + \sqrt{400}}{2} = \frac{4 + 20}{2} = \frac{24}{2} = 12$$

$$y_2 = \frac{4 - \sqrt{400}}{2} = \frac{4 - 20}{2} = \frac{-16}{2} = -8$$

Ответ: $$y_1 = 12, y_2 = -8$$

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г) $$3p^2 + 3 = 10p$$

$$3p^2 - 10p + 3 = 0$$

$$D = (-10)^2 - 4 \cdot 3 \cdot 3 = 100 - 36 = 64$$

$$p_1 = \frac{10 + \sqrt{64}}{6} = \frac{10 + 8}{6} = \frac{18}{6} = 3$$

$$p_2 = \frac{10 - \sqrt{64}}{6} = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}$$

Ответ: $$p_1 = 3, p_2 = \frac{1}{3}$$

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д) $$x^2 - 20x = 20x + 100$$

$$x^2 - 40x - 100 = 0$$

$$D = (-40)^2 - 4 \cdot 1 \cdot (-100) = 1600 + 400 = 2000$$

$$x_1 = \frac{40 + \sqrt{2000}}{2} = \frac{40 + 20\sqrt{5}}{2} = 20 + 10\sqrt{5}$$

$$x_2 = \frac{40 - \sqrt{2000}}{2} = \frac{40 - 20\sqrt{5}}{2} = 20 - 10\sqrt{5}$$

Ответ: $$x_1 = 20 + 10\sqrt{5}, x_2 = 20 - 10\sqrt{5}$$

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е) $$25x^2 - 13x = 10x^2 - 7$$

$$15x^2 - 13x + 7 = 0$$

$$D = (-13)^2 - 4 \cdot 15 \cdot 7 = 169 - 420 = -251$$

D < 0, нет корней

Ответ: нет корней