SOLUTION: Find all solutions of the equation |x^2 - 14x + 29| = 4. Discuss whether or not your solution generates extraneous solutions.

You need to solve actually two quadratic equations

1)  x^2 - 14x + 29 = 4,      (1)     and

2)  x^2 - 14x + 29 = -4      (2)

independently, and then to take the union of their roots as a set of solutions to the absolute value equation.


So, start from eq(1).

x^2 - 14x + 29 = 4  ====>  x^2 - 14x + (29-4) = 0  ====>

x^2 - 14x + 25 = 0  ====>   =  =  =  = .



Now work on eqn(2)

x^2 - 14x + 29 = -4  ====>  x^2 - 14x + (29 +4) = 0  ====>  

x^2 - 14x + 33 = 0  ====>   =  =  = .


So, the solutions to your original equation are these four real numbers:

,  ,    and  .