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Ответ:
Решение:
Решим каждое квадратное уравнение, найдя дискриминант \( D = b^2 - 4ac \) и корни по формуле \( x = \frac{-b ± √{D}}{2a} \).
- \( 3x^2 - 7x + 4 = 0 \). \( D = (-7)^2 - 4 · 3 · 4 = 49 - 48 = 1 \). \( x_1 = \frac{7+1}{6} = \frac{8}{6} = \frac{4}{3} \). \( x_2 = \frac{7-1}{6} = \frac{6}{6} = 1 \).
- \( 5x^2 - 8x + 3 = 0 \). \( D = (-8)^2 - 4 · 5 · 3 = 64 - 60 = 4 \). \( x_1 = \frac{8+2}{10} = 1 \). \( x_2 = \frac{8-2}{10} = \frac{6}{10} = 0.6 \).
- \( 3x^2 - 13x + 14 = 0 \). \( D = (-13)^2 - 4 · 3 · 14 = 169 - 168 = 1 \). \( x_1 = \frac{13+1}{6} = \frac{14}{6} = \frac{7}{3} \). \( x_2 = \frac{13-1}{6} = \frac{12}{6} = 2 \).
- \( 2y^2 - 9y + 10 = 0 \). \( D = (-9)^2 - 4 · 2 · 10 = 81 - 80 = 1 \). \( y_1 = \frac{9+1}{4} = \frac{10}{4} = 2.5 \). \( y_2 = \frac{9-1}{4} = \frac{8}{4} = 2 \).
- \( 5y^2 - 6y + 1 = 0 \). \( D = (-6)^2 - 4 · 5 · 1 = 36 - 20 = 16 \). \( y_1 = \frac{6+4}{10} = 1 \). \( y_2 = \frac{6-4}{10} = \frac{2}{10} = 0.2 \).
- \( 4x^2 + x - 33 = 0 \). \( D = 1^2 - 4 · 4 · (-33) = 1 + 528 = 529 \). \( √{D} = 23 \). \( x_1 = \frac{-1+23}{8} = \frac{22}{8} = 2.75 \). \( x_2 = \frac{-1-23}{8} = \frac{-24}{8} = -3 \).
- \( y^2 - 10y - 24 = 0 \). \( D = (-10)^2 - 4 · 1 · (-24) = 100 + 96 = 196 \). \( √{D} = 14 \). \( y_1 = \frac{10+14}{2} = 12 \). \( y_2 = \frac{10-14}{2} = -2 \).
- \( p^2 + p - 90 = 0 \). \( D = 1^2 - 4 · 1 · (-90) = 1 + 360 = 361 \). \( √{D} = 19 \). \( p_1 = \frac{-1+19}{2} = 9 \). \( p_2 = \frac{-1-19}{2} = -10 \).
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) \( y_1 = 1, y_2 = 0.2 \); е) \( x_1 = 2.75, x_2 = -3 \); ж) \( y_1 = 12, y_2 = -2 \); з) \( p_1 = 9, p_2 = -10 \).
