Question

If \(554_b\) is the base \(b\) representation of the square of the number whose base \(b\) representation is \(24_b,\) then find \(b\).

Answer 1

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

Answer 2

and of course we are to find the b there

Answer 3

I think that looks about right

Answer 4

\[5 \cdot b^2+ 5 \cdot b+4=(2 \cdot b+4)(2 \cdot b +4)\]

Answer 5

this looks like a quadratic equation

Answer 6

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

Answer 7

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

Answer 8

check that middle term and last term you have

Answer 9

They are 16s whoops \(4b2+16b+16\)

Answer 10

\(4b^2\)

Answer 11

right so you have
\[5b^2+5b+4=4b^2+16b+16\]

Answer 12

you need to some subtraction on both sides

Answer 13

okay

Answer 14

so \(b^2=11b+12\)

Answer 15

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

Answer 16

which you can factor the left hand side

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

Right, I remember now

Answer 22

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?

Answer 23

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 )

Answer 24

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

Answer 25

cool stuff :)

Answer 26

Thanks :)