Решение квадратных уравнений: подробный разбор

Ниже представлено подробное решение всех уравнений из таблицы, разбитое по столбцам. Решение уравнений (Столбец 2): 1) \( x^2 + 5x + 4 = 0 \) По теореме Виета: \( x_1 + x_2 = -5 \) \( x_1 \cdot x_2 = 4 \) \( x_1 = -1, x_2 = -4 \) 2) \( 3x^2 + 21x - 90 = 0 \) Разделим на 3: \( x^2 + 7x - 30 = 0 \) \( D = 7^2 - 4 \cdot 1 \cdot (-30) = 49 + 120 = 169 = 13^2 \) \( x = \frac{-7 \pm 13}{2} \) \( x_1 = 3, x_2 = -10 \) 3) \( 4x^2 + 3x - 22 = 0 \) \( D = 3^2 - 4 \cdot 4 \cdot (-22) = 9 + 352 = 361 = 19^2 \) \( x = \frac{-3 \pm 19}{8} \) \( x_1 = 2, x_2 = -2,75 \) 4) \( 6x^2 - 5x - 1 = 0 \) \( D = (-5)^2 - 4 \cdot 6 \cdot (-1) = 25 + 24 = 49 = 7^2 \) \( x = \frac{5 \pm 7}{12} \) \( x_1 = 1, x_2 = -\frac{1}{6} \) 5) \( -x^2 - 3x + 1 = 0 \) \( x^2 + 3x - 1 = 0 \) \( D = 3^2 - 4 \cdot 1 \cdot (-1) = 9 + 4 = 13 \) \( x = \frac{-3 \pm \sqrt{13}}{2} \) 6) \( 3a^2 + a = 7 \Rightarrow 3a^2 + a - 7 = 0 \) \( D = 1^2 - 4 \cdot 3 \cdot (-7) = 1 + 84 = 85 \) \( a = \frac{-1 \pm \sqrt{85}}{6} \) 7) \( 2a^2 - a = 3 \Rightarrow 2a^2 - a - 3 = 0 \) \( D = (-1)^2 - 4 \cdot 2 \cdot (-3) = 1 + 24 = 25 = 5^2 \) \( a = \frac{1 \pm 5}{4} \) \( a_1 = 1,5, a_2 = -1 \) 8) \( 4a^2 - 5 = a \Rightarrow 4a^2 - a - 5 = 0 \) \( D = (-1)^2 - 4 \cdot 4 \cdot (-5) = 1 + 80 = 81 = 9^2 \) \( a = \frac{1 \pm 9}{8} \) \( a_1 = 1,25, a_2 = -1 \) 9) \( 2x^2 + 3x + 1 = 0 \) \( D = 3^2 - 4 \cdot 2 \cdot 1 = 9 - 8 = 1 \) \( x = \frac{-3 \pm 1}{4} \) \( x_1 = -0,5, x_2 = -1 \) Решение уравнений (Столбец 3): 1) \( 3x^2 - 2x - 16 = 0 \) \( D = (-2)^2 - 4 \cdot 3 \cdot (-16) = 4 + 192 = 196 = 14^2 \) \( x = \frac{2 \pm 14}{6} \) \( x_1 = \frac{16}{6} = \frac{8}{3} = 2\frac{2}{3}, x_2 = -2 \) 2) \( -0,5x^2 + 2x - 2 = 0 \) Умножим на -2: \( x^2 - 4x + 4 = 0 \) \( (x - 2)^2 = 0 \) \( x = 2 \) 3) \( x^2 - 6x + 11 = 0 \) \( D = (-6)^2 - 4 \cdot 1 \cdot 11 = 36 - 44 = -8 \) \( D < 0 \), корней нет. 4) \( 5x^2 - 16x + 3 = 0 \) (в таблице опечатка "16", вероятно имелось в виду 16x) \( D = (-16)^2 - 4 \cdot 5 \cdot 3 = 256 - 60 = 196 = 14^2 \) \( x = \frac{16 \pm 14}{10} \) \( x_1 = 3, x_2 = 0,2 \) 5) \( 3x^2 + 4x - 2 = 0 \) \( D = 4^2 - 4 \cdot 3 \cdot (-2) = 16 + 24 = 40 \) \( x = \frac{-4 \pm \sqrt{40}}{6} = \frac{-4 \pm 2\sqrt{10}}{6} = \frac{-2 \pm \sqrt{10}}{3} \) 6) \( 5x^2 - 10x + 6 = 0 \) \( D = (-10)^2 - 4 \cdot 5 \cdot 6 = 100 - 120 = -20 \) \( D < 0 \), корней нет. 7) \( 9y^2 + 6y + 1 = 0 \) \( (3y + 1)^2 = 0 \) \( 3y = -1 \Rightarrow y = -\frac{1}{3} \) 8) \( 5x^2 - 10x + 1 = 0 \) \( D = (-10)^2 - 4 \cdot 5 \cdot 1 = 100 - 20 = 80 \) \( x = \frac{10 \pm \sqrt{80}}{10} = \frac{10 \pm 4\sqrt{5}}{10} = 1 \pm 0,4\sqrt{5} \) 9) \( 3x^2 - 8x + 5 = 0 \) Сумма коэффициентов \( 3 - 8 + 5 = 0 \), значит: \( x_1 = 1, x_2 = \frac{5}{3} = 1\frac{2}{3} \) Решение уравнений (Столбец 4): 1) \( x^2 - 4x + 3 = 0 \Rightarrow x_1 = 1, x_2 = 3 \) 2) \( x^2 + 2x - 3 = 0 \Rightarrow x_1 = 1, x_2 = -3 \) 3) \( x^2 + 3x - 4 = 0 \Rightarrow x_1 = 1, x_2 = -4 \) 4) \( x^2 - x - 12 = 0 \Rightarrow x_1 = 4, x_2 = -3 \) 5) \( x^2 + 8x + 15 = 0 \Rightarrow x_1 = -3, x_2 = -5 \) 6) \( x^2 - 3x - 10 = 0 \Rightarrow x_1 = 5, x_2 = -2 \) 7) \( x^2 - 3x - 18 = 0 \Rightarrow x_1 = 6, x_2 = -3 \) 8) \( x^2 + 5x - 14 = 0 \Rightarrow x_1 = 2, x_2 = -7 \) 9) \( x^2 - 7x + 10 = 0 \Rightarrow x_1 = 2, x_2 = 5 \)