534. Решите уравнение: a) 3x² - 7x + 4 = 0; б) 5x² - 8x + 3 = 0; в) 3х² - 13x + 14 = 0; г) 2y² - 9у + 10 = 0; д) 5y² - 6y + 1 = 0; e) 4x² + x - 33 = 0; ж) у² - 10у - 24 = 0; з) р² + p - 90 = 0.

Решим уравнение:

а) 3x² - 7x + 4 = 0

$$D = (-7)^2 - 4 \cdot 3 \cdot 4 = 49 - 48 = 1$$

$$x_1 = \frac{-(-7) + \sqrt{1}}{2 \cdot 3} = \frac{7+1}{6} = \frac{8}{6} = \frac{4}{3}$$ $$x_2 = \frac{-(-7) - \sqrt{1}}{2 \cdot 3} = \frac{7-1}{6} = \frac{6}{6} = 1$$

б) 5x² - 8x + 3 = 0

$$D = (-8)^2 - 4 \cdot 5 \cdot 3 = 64 - 60 = 4$$

$$x_1 = \frac{-(-8) + \sqrt{4}}{2 \cdot 5} = \frac{8+2}{10} = \frac{10}{10} = 1$$

$$x_2 = \frac{-(-8) - \sqrt{4}}{2 \cdot 5} = \frac{8-2}{10} = \frac{6}{10} = 0,6$$

в) 3х² - 13x + 14 = 0

$$D = (-13)^2 - 4 \cdot 3 \cdot 14 = 169 - 168 = 1$$

$$x_1 = \frac{-(-13) + \sqrt{1}}{2 \cdot 3} = \frac{13+1}{6} = \frac{14}{6} = \frac{7}{3}$$ $$x_2 = \frac{-(-13) - \sqrt{1}}{2 \cdot 3} = \frac{13-1}{6} = \frac{12}{6} = 2$$

г) 2y² - 9у + 10 = 0

$$D = (-9)^2 - 4 \cdot 2 \cdot 10 = 81 - 80 = 1$$

$$y_1 = \frac{-(-9) + \sqrt{1}}{2 \cdot 2} = \frac{9+1}{4} = \frac{10}{4} = 2,5$$ $$y_2 = \frac{-(-9) - \sqrt{1}}{2 \cdot 2} = \frac{9-1}{4} = \frac{8}{4} = 2$$

д) 5y² - 6y + 1 = 0

$$D = (-6)^2 - 4 \cdot 5 \cdot 1 = 36 - 20 = 16$$

$$y_1 = \frac{-(-6) + \sqrt{16}}{2 \cdot 5} = \frac{6+4}{10} = \frac{10}{10} = 1$$ $$y_2 = \frac{-(-6) - \sqrt{16}}{2 \cdot 5} = \frac{6-4}{10} = \frac{2}{10} = 0,2$$

e) 4x² + x - 33 = 0

$$D = 1^2 - 4 \cdot 4 \cdot (-33) = 1 + 528 = 529$$

$$x_1 = \frac{-1 + \sqrt{529}}{2 \cdot 4} = \frac{-1+23}{8} = \frac{22}{8} = 2,75$$ $$x_2 = \frac{-1 - \sqrt{529}}{2 \cdot 4} = \frac{-1-23}{8} = \frac{-24}{8} = -3$$

ж) у² - 10у - 24 = 0

$$D = (-10)^2 - 4 \cdot 1 \cdot (-24) = 100 + 96 = 196$$

$$y_1 = \frac{-(-10) + \sqrt{196}}{2 \cdot 1} = \frac{10+14}{2} = \frac{24}{2} = 12$$ $$y_2 = \frac{-(-10) - \sqrt{196}}{2 \cdot 1} = \frac{10-14}{2} = \frac{-4}{2} = -2$$

з) р² + p - 90 = 0

$$D = 1^2 - 4 \cdot 1 \cdot (-90) = 1 + 360 = 361$$

$$p_1 = \frac{-1 + \sqrt{361}}{2 \cdot 1} = \frac{-1+19}{2} = \frac{18}{2} = 9$$ $$p_2 = \frac{-1 - \sqrt{361}}{2 \cdot 1} = \frac{-1-19}{2} = \frac{-20}{2} = -10$$

Ответ: а) $$x_1 = \frac{4}{3}$$, $$x_2 = 1$$; б) $$x_1 = 1$$, $$x_2 = 0,6$$; в) $$x_1 = \frac{7}{3}$$, $$x_2 = 2$$; г) $$y_1 = 2,5$$, $$y_2 = 2$$; д) $$y_1 = 1$$, $$y_2 = 0,2$$; e) $$x_1 = 2,75$$, $$x_2 = -3$$; ж) $$y_1 = 12$$, $$y_2 = -2$$; з) $$p_1 = 9$$, $$p_2 = -10$$.