Question

if a + b=2 and a^2+ b^2= 5 then what is the value of a^3 + b^3?

Answer 1

i think you are going to use \(a^3+b^3)=(a+b)(a^2+ab+b^2)\)
at the moment you know \((a+b)\) and \(a^2+b^2\) so all that is left to know is \(ab\)
any idea how to find it?

Answer 2

i made a mistake there, it is
\[a^3+b^3=(a+b)(a^2-ab+b^2)\] but the idea is the same, you need \(ab\)

Answer 3

if you want a hint, to find \(ab\) note that \((a+b)^2=a^2+b^2+2ab\) you know all of those numbers except \(2ab\)

Answer 4

wait? can you explain again? do i solve for a and b using the other formulas?

Answer 5

you know that \(a+b=4\), and that \(a^2+b^2=5\)

Answer 6

and also that \(a^3+b^3=(a+b)(a^2-ab+b^2)\)
so \(a^3+b^3=4\times (5-ab)\)

Answer 7

so far so good?

Answer 8

you cannot solve for \(a\) and for \(b\) but you do not need to. you need to solve for \(ab\)

Answer 9

it is not 4, it is 2

Answer 10

ok my fault.
but no matter, you have
\[a^3+b^3=2(5-ab)\]

Answer 11

what you need to finish is \(ab\)

Answer 12

since \((a+b)^2=a^2+b^2+ab\) you have \(2^2=5+ab\)

Answer 13

so would it be 5=2(5-ab)?

Answer 14

no the left hand side you do not know
it is the \(a^3+b^3\) that you are looking for

Answer 15

it is
\[a^3+b^3=2(5-ab)\]

Answer 16

never mind....yea i just saw what i was doing...

Answer 17

so since \((a+b)^2=a^2+b^2+2ab\) you have
\[2^2=5+2ab\]
\[4=5+2ab\]
\[2ab=-1\] and so \(ab=-\frac{1}{2}\)

Answer 18

making
\[a^3+b^3=2(5+\frac{1}{2})\]

Answer 19

(a+b)^2=a^2+b^2+2ab?

Answer 20

yes

Answer 21

I see where you got it from.

Answer 22

it is usually written as
\[(a+b)^2=a^2+2ab+b^2\] but i wrote it differently as
\[(a+b)^2=a^2+b^2+2ab\] so i could replace the \(a^2+b^2\) by 5

Answer 23

but wouldnt it be -ab?

Answer 24

would what be \(-ab\) ?

Answer 25

oh you need \(-ab\) for sure, so answer your question
but first we find \(2ab=-1\) and so \(-ab=\frac{1}{2}\)

Answer 26

when you said it is usually written as
(a+b)2=a2+2ab+b2
but i wrote it differently as
(a+b)2=a2+b2+2ab

Answer 27

yes that is correct

Answer 28

it is not \(-ab\)
\[(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2\]

Answer 29

but where in this process does a^3 + b^3 go to?

Answer 30

i think i have confused you

Answer 31

we start with factoring
\[a^3+b^3=(a+b)(a^2-ab+b^2)\]

Answer 32

you want to know what that is and so far we know \(a+b=2\) and \(a^2+b^2=5\) so we have most of it, we are just missing one piece
we have
\[a^3+b^3=2\times (5-ab)\]

Answer 33

the only part that we don't know is \(-ab\)

Answer 34

to find \(-ab\) we look at
\[(a+b)^2\] which is \(2^2=4\) and notice that
\[(a+b)^2=4=a^2+b^2+2ab\]
we also know \(a^2+b^2=5\) so the above equation becomes
\[4=5+2ab\]

Answer 35

i see what you are saying but the only thing is where you get the (a+B)^2

Answer 36

oh it was a gimmick to find \(-ab\) that is all
a trick
in other words i thought it up
you have to find \(ab\) somehow

Answer 37

all you know is \(a^2+b^2=5\) and \(a+b=2\) so you have to use what you got to get what you want

Answer 38

Okay! It makes so much more sense now! Thanks!!!

Answer 39

yw