The discriminant formula.
b2 - 4ac
32 - 4(1)(8)
9 - 32 = - 23
===========This shows no real roots to this function.
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x = -5 x = 2
if x2 + 7 = 37, then x2 = 29 and x = ±√29
3x3 + 6x2 - 24x = 3x(x2 + 2x - 8) = 3x(x2 + 4x - 2x - 8) = 3x[ x(x + 4) - 2(x + 4) ] = 3x(x - 2)(x + 4)
x2/(3x2 - 5x - 2) - [2x/ (3x + 1)][1/(x - 2)] = x2/(3x2 - 6x + x - 2) - 2x/(3x + 1)(x - 2) = x2/[3x(x - 2) + (x - 2)] - 2x/(3x + 1)(x - 2) = x2/(3x + 1)(x - 2) - 2x/(3x + 1)(x - 2) = (x2 - 2x)/(3x + 1)(x - 2) = x(x - 2)/(3x + 1)(x - 2) = x/(3x + 1)
x^2 + 3x +7 = 0 doesn't factor neatly. Applying the quadratic equation, we find two imaginary solutions: (-3 plus or minus the square root of -19) divided by 2x = -1.5 + 2.179449471770337ix = -1.5 - 2.179449471770337iwhere i is the square root of negative 1