## h0pe one year ago If $$554_b$$ is the base $$b$$ representation of the square of the number whose base $$b$$ representation is $$24_b,$$ then find $$b$$.

1. myininaya

wait so it kind of looks like it is saying this: $554_b=(24_b) \cdot (24_b)$

2. myininaya

and of course we are to find the b there

3. h0pe

I think that looks about right

4. myininaya

$5 \cdot b^2+ 5 \cdot b+4=(2 \cdot b+4)(2 \cdot b +4)$

5. myininaya

this looks like a quadratic equation

6. h0pe

so then $$(2b+4)(2b+4)$$ is $$4b^2+8b+8$$ right?

7. myininaya

well not exactly 4(4)=16 and 2b(4)+4(2b)=16b

8. myininaya

check that middle term and last term you have

9. h0pe

They are 16s whoops $$4b2+16b+16$$

10. h0pe

$$4b^2$$

11. myininaya

right so you have $5b^2+5b+4=4b^2+16b+16$

12. myininaya

you need to some subtraction on both sides

13. h0pe

okay

14. h0pe

so $$b^2=11b+12$$

15. myininaya

$\text{ or } b^2-11b-12=0$

16. myininaya

which you can factor the left hand side

17. myininaya

or you can use the quadratic formula if you really want to but factoring the left hand expression is not too bad

18. myininaya

you will get one b that makes sense and the other b that makes no sense

19. h0pe

what do you mean by factoring? sorry it's been a while since I've used that. 0.0

20. myininaya

like to factor something like x^2-5x+6 you can look for two numbers that multiply to be 6 and add up to be -5 so -2(-3)=-6 and -2+(-3)=-5 so x^2-5x+6 can be written as (x-2)(x-3) and if you wanted to solve x^2-5x+6=0 you can write this first as (x-2)(x-3)=0 then set both factors equal to zero x-2=0 or x-3=0 so x=2 or x=3 you can find two numbers that multiply to be -12 and add up to be -11

21. h0pe

Right, I remember now

22. h0pe

For the case of $$b^2−11b−12=0$$ it would be -12 and 1, wouldn't it be? So the equation would be $$(b-12)(b+1)$$ right?

23. myininaya

bingo so you have to solve (b-12)(b+1)=0 which means you have either b-12=0 or b+1=0 (or both <--both will not work in this case and I hope you will see why )

24. h0pe

The answer has to be $$b=12$$, because there aren't negative bases.

25. myininaya

cool stuff :)

26. h0pe

Thanks :)