No real solutions exist for the given equations, as the square roots of the resulting constants are imaginary. The impossibility arises from attempting to find real numbers whose squares equal the specified negative values.
Let's solve each equation using the square roots method:
14. (x+4)^2 = -90
(x + 4)^2 = -90
Take the square root of both sides: x + 4 = ±√(-90)
Notice that √(-90) is imaginary, so there are no real solutions for this equation.
The square of any real number is non-negative, so there's no real number whose square is -90.
16. -(x-5)^2 = 108
-(x - 5)^2 = 108
x - 5 = ±√(-108)
Again, √(-108) is imaginary, so there are no real solutions.
18. 5(x-3)^2 = -225
5(x - 3)^2 = -225
(x - 3)^2 = -45
x - 3 = ±√(-45)
√(-45) is imaginary, so there are no real solutions.
20. -5/2(x+1)^2 = 30
-5/2(x + 1)^2 = 30
(x + 1)^2 = -12
x + 1 = ±√(-12)
√(-12) is imaginary, so there are no real solutions.
All four equations have no real solutions, as the square roots of the resulting constants are imaginary.