a)
9 + 6x^2 - 8x + 13 = 2(7x + 5)
6x^2 - 8x + 22 = 14x + 10
6x^2 - 8x - 14x + 22 - 10 = 0
6x^2 - 22x + 12 = 0
3x^2 - 11x + 6 = 0
3x^2 - 2x - 9x + 6 = 0
(3x^2 - 2x) - (9x - 6) = 0
x(3x - 2) - 3(3x - 2) = 0
(3x - 2)(x - 3) = 0
3x - 2 = 0 or x - 3 = 0
x = 2/3 or x = 3
b)
wx^4 - 4x^3 = x^2
wx^4 = 4x^3 + x^2
w = (4x^3 + x^2) / x^4
w = (4x + 1) / x^2
c)
2x^2 - 5x + 6 = 0
(2x^2 - 5x) + 6 = 0
2(x^2 - (5/2)x) + 6 = 0
2(x - (5/4))^2 - (25/18) + (108/18) = 0
2(x - (5/4))^2 + (83/18) = 0
2(x - (5/4))^2 = -83/18
(x - (5/4))^2 = -83/36
x - 5/4 = +/- sqrt(-83/36)
x = 5/4 + sqrt(-83/36) or x = 5/4 - sqrt(-93/36)
Note that the answer is complex.
d)
4x + 3x^2 - 11 = 0
3x^2 + 4x - 11 = 0
(3x^2 + 4x) - 11 = 0
3(x^2 + (4/3)x) - 11 = 0
3(x + (2/3))^2 - 4/3 - 33/3 = 0
3(x + (2/3))^2 - 37/3 = 0
3(x + (2/3))^2 = 37/3
(x + (2/3))^2 = 37 / 9
x + 2/3 = +/-sqrt(37 / 9)
x = -2/3 + sqrt(37/9) or x = -2/3 - sqrt(37/9)