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

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lemjay | High School Teacher | (Level 3) Senior Educator

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`|x^2-14x+29|=4`

Take note that if we take absolute value of a number, the result is always positive (|n| >= 0). Like |7| = 7  and  |-7| = 7.

Applying this property to our equation, we have to set it to:

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

Then, solve the equation individually.

For the first equation, the values of x are:

`x^2 - 14x + 29 = 4`

`x^2-14x+ 29 - 4 = 0`

`x^2-14x+25 = 0`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(-(-14)+-sqrt((-14)^2-4(1)(25)))/((2)(1)) =(14+-sqrt96)/2`

`x=2.1 , 11.9 `

For the second equation, the values of x are:

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

`x^2 -14x + 29 = -4`

`x^2 - 14x + 29 + 4=0`

`x^2 - 14x + 33 = 0`

`x=(-(-14)+-sqrt((-14)^2-4(1)(33)))/((2)(1)) = (14+-sqrt64)/2=(14+-8)/2`

`x=3,11`

And, take the union of the values of x in our two equations.

Therefore, the solution is  `x = {2.1 , 3 , 11, 11.9}` .

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