Solving problem by finding factor

[dsc2010]

Hello all,
am having some trouble find the right rule/logic to use to solve the following question.
am guessing i should assume x2 => 10 000 and x2 - 9 = 10000 -9 , therefore 9991 but cant put my finger on the right way to do it.
Am more interested about the how the second part of the question is connected to the first 1.

Thanks in advance.

Question:



Working:




Answer:

 

[Zermelo]

9991 = 10000-9, right? Now you got something similar to (1), where x = 100.

 

[pka]

Hello all,
am having some trouble find the right rule/logic to use to solve the following question.
am guessing i should assume x2 => 10 000 and x2 - 9 = 10000 -9 , therefore 9991 but cant put my finger on the right way to do it.
Am more interested about the how the second part of the question is connected to the first 1.

\(9991=1000-9=10^4-3^2=(10^2-3)(10^2+3)=97\cdot 103\)

 

[dsc2010]

\(9991=1000-9=10^4-3^2=(10^2-3)(10^2+3)=97\cdot 103\)

Thanks pka
i reached the same conclusion
the reason i wanted to know the how is so that i can share with my kid how to present this in an exam
specially the part : (10² -3) (10² +3)

would the following be enough or do i have to specify an assumption or rule :
10 000 - 9 = 9991

from part (a):
x² - 9 = 9991

let x² = 10 000
x = root of 10 000 = 100
therefore
100² - 9 = 9991
(10² -3) (10² +3) = 9991
factors :
(10² -3) = 100 -3 = 97 <=
(10² +3) = 100 +3 =103 <=

??