SOLUTION: Find all solutions of the equation |x^2 - 14x + 29| = 4.

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Question 1044821: Find all solutions of the equation |x^2 - 14x + 29| = 4.
Found 2 solutions by ikleyn, Boreal:
Answer by ikleyn(52810)   (Show Source): You can put this solution on YOUR website!
.
Find all solutions of the equation |x^2 - 14x + 29| = 4.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This equation deploys in two independent equations: 

1)   =   --->   =   --->   = ;

                                                        = ,   = . 

and 

2)   =   --->   =  ---> (x-11)*(x-3) = 0  --->  x = 11  or  x = 3.


Answer.   = ,   = ,   = 11  and   = 3.




Plots y = |x^2 - 14x + 29| and y = 4.

See also the lessons
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3
    - HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 1
    - HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 2
in this site.


Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
|x^2 - 14x + 29| = 4.
start with x^2-14x+29=4
x^2-14x+25=0
x=(1/2)(14+/- sqrt (196-100); sqrt 96=4 sqrt (6)
x=7+2 sqrt (6) and 7-2 sqrt (6). Numerically, these are about 11.9 and 2.1
Check and 11.9 and 2.1 are both within 0.01 (rounding error).

The other solution is when x^2-14x+29=-4, because |-4|=4.
Now x^2-14x+33=0
(x-11)(x-3)=0
x=11,3
Check in the original |121-154+29|=|-4)=4; 9-42+29=|-4|=4

Exact solutions are 7-2 sqrt(6),3,11, and 7+2 sqrt (6)

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