Заполните таблицу по теореме Виета для следующих квадратных уравнений: 1) x² + 3x + 1 = 0 2) x² - 2x - 5 = 0 3) x² + 4x - 3 = 0 4) x² - 4x + 3 = 0 5) x² + 3x - 3 = 0 6) x² + x - 1 = 0 7) x² - 2x = 0 8) x² - 7 = 0 9) x² + 5x + 8 = 0 10) x² + 6x + 9 = 0 11) x² - 12x + 36 = 0 12) x² + 3x + 9 = 0 13) x² - x - 2 = 0 14) x² - 4x + 4 = 0 15) 10x + x² - 25 = 0

Привет, ребята! Сейчас мы решим эти уравнения, используя теорему Виета. Наша задача - вычислить дискриминант, сумму и произведение корней для каждого уравнения. Напомню, что для квадратного уравнения вида (x^2 + px + q = 0), если (x_1) и (x_2) – корни уравнения, то: * (x_1 + x_2 = -p) * (x_1 cdot x_2 = q) А дискриминант (D = p^2 - 4q). Теперь приступим к решению: 1) (x^2 + 3x + 1 = 0) * (p = 3), (q = 1) * (D = 3^2 - 4 cdot 1 = 9 - 4 = 5) * (x_1 + x_2 = -3) * (x_1 cdot x_2 = 1) 2) (x^2 - 2x - 5 = 0) * (p = -2), (q = -5) * (D = (-2)^2 - 4 cdot (-5) = 4 + 20 = 24) * (x_1 + x_2 = 2) * (x_1 cdot x_2 = -5) 3) (x^2 + 4x - 3 = 0) * (p = 4), (q = -3) * (D = 4^2 - 4 cdot (-3) = 16 + 12 = 28) * (x_1 + x_2 = -4) * (x_1 cdot x_2 = -3) 4) (x^2 - 4x + 3 = 0) * (p = -4), (q = 3) * (D = (-4)^2 - 4 cdot 3 = 16 - 12 = 4) * (x_1 + x_2 = 4) * (x_1 cdot x_2 = 3) 5) (x^2 + 3x - 3 = 0) * (p = 3), (q = -3) * (D = 3^2 - 4 cdot (-3) = 9 + 12 = 21) * (x_1 + x_2 = -3) * (x_1 cdot x_2 = -3) 6) (x^2 + x - 1 = 0) * (p = 1), (q = -1) * (D = 1^2 - 4 cdot (-1) = 1 + 4 = 5) * (x_1 + x_2 = -1) * (x_1 cdot x_2 = -1) 7) (x^2 - 2x = 0) * (p = -2), (q = 0) * (D = (-2)^2 - 4 cdot 0 = 4) * (x_1 + x_2 = 2) * (x_1 cdot x_2 = 0) 8) (x^2 - 7 = 0) * (p = 0), (q = -7) * (D = 0^2 - 4 cdot (-7) = 28) * (x_1 + x_2 = 0) * (x_1 cdot x_2 = -7) 9) (x^2 + 5x + 8 = 0) * (p = 5), (q = 8) * (D = 5^2 - 4 cdot 8 = 25 - 32 = -7) * (x_1 + x_2 = -5) * (x_1 cdot x_2 = 8) 10) (x^2 + 6x + 9 = 0) * (p = 6), (q = 9) * (D = 6^2 - 4 cdot 9 = 36 - 36 = 0) * (x_1 + x_2 = -6) * (x_1 cdot x_2 = 9) 11) (x^2 - 12x + 36 = 0) * (p = -12), (q = 36) * (D = (-12)^2 - 4 cdot 36 = 144 - 144 = 0) * (x_1 + x_2 = 12) * (x_1 cdot x_2 = 36) 12) (x^2 + 3x + 9 = 0) * (p = 3), (q = 9) * (D = 3^2 - 4 cdot 9 = 9 - 36 = -27) * (x_1 + x_2 = -3) * (x_1 cdot x_2 = 9) 13) (x^2 - x - 2 = 0) * (p = -1), (q = -2) * (D = (-1)^2 - 4 cdot (-2) = 1 + 8 = 9) * (x_1 + x_2 = 1) * (x_1 cdot x_2 = -2) 14) (x^2 - 4x + 4 = 0) * (p = -4), (q = 4) * (D = (-4)^2 - 4 cdot 4 = 16 - 16 = 0) * (x_1 + x_2 = 4) * (x_1 cdot x_2 = 4) 15) (x^2 + 10x - 25 = 0) * (p = 10), (q = -25) * (D = 10^2 - 4 cdot (-25) = 100 + 100 = 200) * (x_1 + x_2 = -10) * (x_1 cdot x_2 = -25) Заполненная таблица: | Уравнение | D | x1 + x2 | x1 * x2 | Знак дискриминанта | | ------------------- | ------ | ------- | ------- | -------------------- | | x² + 3x + 1 = 0 | 5 | -3 | 1 | > 0 | | x² - 2x - 5 = 0 | 24 | 2 | -5 | > 0 | | x² + 4x - 3 = 0 | 28 | -4 | -3 | > 0 | | x² - 4x + 3 = 0 | 4 | 4 | 3 | > 0 | | x² + 3x - 3 = 0 | 21 | -3 | -3 | > 0 | | x² + x - 1 = 0 | 5 | -1 | -1 | > 0 | | x² - 2x = 0 | 4 | 2 | 0 | > 0 | | x² - 7 = 0 | 28 | 0 | -7 | > 0 | | x² + 5x + 8 = 0 | -7 | -5 | 8 | < 0 | | x² + 6x + 9 = 0 | 0 | -6 | 9 | = 0 | | x² - 12x + 36 = 0 | 0 | 12 | 36 | = 0 | | x² + 3x + 9 = 0 | -27 | -3 | 9 | < 0 | | x² - x - 2 = 0 | 9 | 1 | -2 | > 0 | | x² - 4x + 4 = 0 | 0 | 4 | 4 | = 0 | | x² + 10x - 25 = 0 | 200 | -10 | -25 | > 0 |