Question 1162005
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<pre>

In positive numbers, the solution is x = 12:


    12^2 + (12-7)^2 = 144 + 25 = 169 = 13^2.


The triple (5,12,13) is so called Pythagorean triple - next after the Pythagorean triple (3,4,5).


In negative numbers, the solution is x = -5; then


    (-5)^2 + (-5-7)^2 = 25 + 144 = 169 = 13^2.


You can solve the original equation formally


    x^2 + x^2 - 14x + 49 = 169

    2x^2 - 14x - 120 = 0

     x^2 - 7x  -  60 = 0      (*)


Next, factor the left side


    (x-12)*(x + 5) = 0,


giving you the same two roots  x= 12  and  x= -5.


Or, alternatively, you can apply the quadratic formula to solve equation (*).
</pre>

Solved and explained.


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On quadratic formula, see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Introduction-Into-Quadratics.lesson>Introduction into Quadratic Equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/proof-of-quadratic-by-completing-the-square.lesson>PROOF of quadratic formula by completing the square</A>

in this site.



On Pythagorean triples, see this Wikipedia articles

https://en.wikipedia.org/wiki/Pythagorean_triple#:~:text=A%20Pythagorean%20triple%20consists%20of,%2B%20b2%20%3D%20c2.&text=The%20name%20is%20derived%20from,lengths%20of%20a%20right%20triangle.