3h^2 + 2h - 16 = 0
Using the Quadratic Formula, we find:
h = (-2 ± √(2^2 - 4(3)(-16))) / (2 * 3)
h = (-2 ± √(4 + 192)) / 6
h = (-2 ± √196) / 6
h = (-2 ± 14) / 6
h = (12, -6) / 6
h = 2, -1
So the solutions to the equation are h = 2 and h = -1.
8q^2 - 10q + 3 = 0
Using the Quadratic Formula, we find:
q = (-(-10) ± √((-10)^2 - 4(8)(3))) / (2 * 8)
q = (10 ± √(100 - 96)) / 16
q = (10 ± √4) / 16
q = (10 ± 2) / 16
q = (12, 8) / 16
q = 3/2, 1/2
So the solutions to the equation are q = 3/2 and q = 1/2.
10r^2 - 21r = -4r + 6
Combining like terms, we find:
10r^2 - 25r + 6 = 0
Using the Quadratic Formula, we find:
r = (-(-25) ± √((-25)^2 - 4(10)(6))) / (2 * 10)
r = (25 ± √(625 - 240)) / 20
r = (25 ± √385) / 20
r = (25 ± 19.7228) / 20
r = (44.7228, 5.2772) / 20
r = 2.2362, 0.2636
So the solutions to the equation are r = 2.2362 and r = 0.2636.
12k^2 + 15k = 16k + 20
Combining like terms, we find:
12k^2 - k = 16k + 20
12k^2 - 17k = 20
Using the Quadratic Formula, we find:
k = (-(-17) ± √((-17)^2 - 4(12)(20))) / (2 * 12)
k = (17 ± √(289 - 960)) / 24
k = (17 ± √(-671)) / 24
k = (17 ± NaN) / 24
Since the square root of a negative number is not defined, there are no real solutions to the equation.
So the solutions to the equation are (x,b) = (2.2362,0.2636), (3/2,1/2), (2,-1).