## aroub 3 years ago 14- Find a decimal approximation to the nearest tenth: i) √82 Is there ANY easy way other than: Find the perfect squares that is before and after the number 82 and then add the both numbers and blah..blah..

1. asnaseer

since it is to the "nearest tenth" you won't have many numbers that you need to try. start with 9 since 9*9=81 and increase by one tenth until you get an answer that is closest to 82.

2. aroub

Can you show me?

3. asnaseer

another possible method might be to use the fact that:\[(a+b)^2=a^2+2ab+b^2\]so set a=9 and you need to find b such that:\[(9+b)^2=82\]

4. asnaseer

therefore:\[(9+b)^2=81+2*9*b+b^2=81+18b+b^2\]therefore:\[81+18b+b^2=82\]therefore:\[b^2+18b=1\]then try b=0.1, 0.2, ....

5. aroub

Oh cool =) Can you do this method if they asked you " Find a decimal approximation to the nearest hundredth" ? @asnaseer

6. asnaseer

yes, then you would need to try b=0.01, 0.02, etc. However, this might take too long, so a better strategy in this case might be to use a technique where you keep halving the interval, e.g.: try b=0.50 ---> if this is too low, then try b=0.75 ---> otherwise try b=0.25 that will converge quicker.

7. asnaseer

have you heard of the Newton-Raphson method?

8. asnaseer

if you have, then that will converge to the answer faster.

9. aroub

Actually no I havent

10. asnaseer

no worries - just use one (or more) of the techniques mentioned above to see which one you prefer.

11. aroub

Thank you so much =D

12. asnaseer

yw :)

13. aroub

Okay, I got a bit stuck using your method.. My method is SOOOOO complicated. Anyway, right you said you try b=0.1,0.2.. like this: b^2+18b=1 0.1^2=18(0.1)=1 ? and if it didnt work you try 0.2 and if didnt work you try 0.3 and so on? O.o

14. aroub

@asnaseer :)

15. asnaseer

you won't find an exact answer - you are supposed to find the nearest answer to one tenth

16. aroub

Yeah and I think you have to round it at the end right?

17. asnaseer

ok, let me try and explain. we said we can re-arrange the question to this form:\[b^2+18b=1\]

18. aroub

but my question is you substitute b with 0.1 like that? until you find an answer to the nearest tenth?

19. asnaseer

so what we need to do is to find what value of b will give an answer that is "closest" to 1. we can re-arrange further to get:\[b^2+18b-1=0\]so now we need to find a value for b that gives the closest answer to zero. so, lets try b=0.0, we get \(b^2+18b-1=-1\) so, lets try b=0.1, we get \(b^2+18b-1=0.81\) so, lets try b=0.2, we get \(b^2+18b-1=2.64\) so the "closest" answer is when b=0.1 therefore, the answer is 9.1 to nearest tenth.

20. asnaseer

does that make sense?

21. aroub

It makes sense till so the "closest" answer is when b=0.1 and then how did it become 9.1?

22. asnaseer

because we started this from:\[(9+b)^2=82\]now we have found 'b', answer is '9+b'

23. asnaseer

remember we said \(9^2=81\) and we are trying to find a number that, when squared equals 82. so that number must be 9 plus a little bit. we called that "little bit" 'b'

24. aroub

This is even more complicated lol but yeah i understood how =) Thank youu!!

25. asnaseer

Afwan :)