Final answer:
To solve the given quadratic equations, we can use the quadratic formula. The solutions for each equation are: a) x = -1, -3/3; b) x = -2, -7/4; c) x = 1, 3/4; d) x = -1/4, 1/4; e) x = 0, 0; f) x = 3/2, 3/2.
Step-by-step explanation:
To solve the given quadratic equations, we can use the quadratic formula.
a) For the equation 3x² + 28x + 9 = 0:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (-28 ± √(28² - 4(3)(9))) / (2(3))
x = (-28 ± √(784 - 108)) / 6
x = (-28 ± √676) / 6
x = (-28 ± 26) / 6
x = -1, -3/3
b) For the equation 4k² + 19k + 15 = 0:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (-19 ± √(19² - 4(4)(15))) / (2(4))
x = (-19 ± √(361 - 240)) / 8
x = (-19 ± √121) / 8
x = (-19 ± 11) / 8
x = -2, -7/4
c) For the equation 8y² - 22y + 15 = 0:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (-(-22) ± √((-22)² - 4(8)(15))) / (2(8))
x = (22 ± √(484 - 480)) / 16
x = (22 ± √4) / 16
x = (22 ± 2) / 16
x = 1, 3/4
d) For the equation 16b² - 1 = 0:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (0 ± √(0² - 4(16)(-1))) / (2(16))
x = (0 ± √(0 + 64)) / 32
x = (0 ± √64) / 32
x = (0 ± 8) / 32
x = -1/4, 1/4
e) For the equation 10m² + 30m = 0:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (0 ± √(0² - 4(10)(0))) / (2(10))
x = (0 ± √0) / 20
x = (0 ± 0) / 20
x = 0, 0
f) For the equation 4x² - 12x + 9 = 0:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (12 ± √((-12)² - 4(4)(9))) / (2(4))
x = (12 ± √(144 - 144)) / 8
x = (12 ± √0) / 8
x = (12 ± 0) / 8
x = 3/2, 3/2