Question 1092338
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Find all solutions of the equation |x^2 - 14x + 29| = 4. Discuss whether or not your solution generates extraneous solutions.
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<pre>
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  ====>  {{{x[1,2]}}} = {{{(14 +- sqrt(14^2 - 4*25))/2}}} = {{{(14 +- sqrt(96))/2}}} = {{{7 +- sqrt(24)}}} = {{{7 +- 2*sqrt(6)}}}.



Now work on eqn(2)

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

x^2 - 14x + 33 = 0  ====>  {{{x[3,4]}}} = {{{(14 +- sqrt(14^2 +4*33))/2}}} = {{{(14 +- sqrt(328))/2}}} = {{{7 +- sqrt(82)}}}.


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

{{{7 + 2*sqrt(6)}}},  {{{7 - 2*sqrt(6)}}},  {{{7 + sqrt(82)}}}  and  {{{7 - sqrt(82)}}}.
</pre>


{{{graph( 500, 500, -5.5, 15.5, -5.5, 25.5,
          abs(x^2 - 14x + 29), 4
)}}}


Plot y = {{{abs(x^2 - 14x + 29)}}}  and y = 4.



These four solutions are x-coordinates of intersection points.


My solution did not produce extraneous solutions.



---------------
On solving absolute value equations see the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/-Absolute-Value-equations.lesson>Absolute Value equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-1.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-2.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-3.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Quadratic-Terms-under-Abs-Value-sign-L-1.lesson>HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Quadratic-Terms-under-Abs-Value-sign-L-2.lesson>HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 2</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic
"<U>Solving Absolute values equations</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I

https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.