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В решении.
Объяснение:
Решить неравенства:
г) -5х² - 11х - 6 >= 0
Приравнять к нулю и решить как квадратное уравнение:
-5х² - 11х - 6 = 0/-1
5х² + 11х + 6 = 0
D=b²-4ac =121 - 120 = 1 √D=1
х₁=(-b-√D)/2a
х₁=(-11-1)/10
х₁= -12/10
х₁= -1,2;
х₂=(-b+√D)/2a
х₂=(-11+1)/10
х₂= -10/10
х₂= -1.
Теперь начертить СХЕМУ параболы (ничего вычислять не нужно), которую выражает данное уравнение, ветви направлены вниз, парабола пересекает ось Ох при х= -1,2 и х= -1, отметить эти точки схематично, смотрим на график.
По графику ясно видно, что у >= 0 (как в неравенстве) при значениях х от -1,2 до х= -1.
Решение неравенства: х∈ [-1,2; -1].
Неравенство нестрогое, скобки квадратные.
д) 9х² - 12х + 4 > 0
Приравнять к нулю и решить как квадратное уравнение:
9х² - 12х + 4 = 0
D=b²-4ac =144 - 144 = 0 √D=0
х=(-b±√D)/2a
х=(12±0)/18
х=12/18
х=2/3.
Уравнение имеет один корень, это означает, что парабола не пересекает ось Ох, а "стоит" на ней, ветви направлены вверх.
у > 0 (как в неравенстве) влево и вправо от значения х = 2/3.
Решение неравенства (-∞; 2/3)∪(2/3; +∞).
Неравенство строгое, скобки круглые.
е) 4х² - 12х + 9 <= 0
Приравнять к нулю и решить как квадратное уравнение:
D=b²-4ac =144 - 144 = 0 √D=0
х=(-b±√D)/2a
х=(12±0)/8
х=12/8
х=1,5.
Уравнение имеет один корень, это означает, что парабола не пересекает ось Ох, а "стоит" на ней, ветви направлены вверх.
Находим по графику значения х, при которых у<=0 (как в неравенстве).
у < 0 не существует, так как парабола выше оси Ох, а у = 0 при х = 1,5.
Решение неравенства {1,5}.
Решение единичное, скобки фигурные.