Question

Help me prove that these two expressions are equal.

Answer 1

I'm doing mathematical induction in honors pre-calc, and i just can't do the factoring.

Answer 2

I need to prove that \[(k+1)^{2}(k+2)^{2}\] is equal to \[k ^{2}(k+1)^{2}+4(k+1)^{3}\]

Answer 3

tricky factoring yeah
simply write \((k+2)^2 = k^2+4(k+1) \)

Answer 4

okay... my professor does this weird thing where he takes out a k from the 2nd and 3rd parts but i dont know if i have to do that

Answer 5

that's cute!

Answer 6

also, what happened to the 4(k+1)^{3} power?

Answer 7

\[(k+1)^2 \color{red}{(k+2)^2} =(k+1)^2 \color{red}{(k^2+4(k+1))} = (k+1)^2\color{red}{k^2} + (k+1)^2 \color{red}{4(k+1)}\]

Answer 8

in my original equation it is\[+ 4(k+1)^{3}\]

Answer 9

not just to the first power

Answer 10

i feel like those two couldnt ever be

Answer 11

equal.. do you know how to do mathmatical induction? could i show you the problem?

Answer 12

maybe i messed up somewhere else

Answer 13

nope we're almost done, just stare at the second term :
\[(k+1)^2 \color{red}{4(k+1)}\]

Answer 14

use the exponent property \( a^2\cdot a = a^3\)

Answer 15

\[(k+1)^2 \color{red}{4(k+1)} = 4(k+1)^2 \color{red}{(k+1)} = 4(k+1)^3 \]

Answer 16

yes i understand that, but i'm not sure how that helps me...

Answer 17

ohhhh wait a second

Answer 18

ok waiting :)

Answer 19

Both expression expand to:
\[k^4+6 k^3+13 k^2+12 k+4 \]

Answer 20

alright. so we got rid of (k+1) squared on both sides, twice on the second side.
so i'm left with \[(k+2)^{2}=k ^{2}+4(k+1)\]
and i can just simplify to \[k^{2}+4k+4\] on both sides... but i still have to prove that it's equal to the original

Answer 21

or do i not have to, because i technically simplified the original one too?

Answer 22

Notice we have started with left hand side, manipulated it and arrived at right hand side

Answer 23

yeah okay... one second

Answer 24

this isn't the same problem but a similar one that my professor did... could we try it this way?

Answer 25

i really just don't understand what he did

Answer 26

\[\begin{align}\text{left hand side }&= (k+1)^2 \color{red}{(k+2)^2} \\&=(k+1)^2 \color{red}{(k^2+4(k+1))} \\&= (k+1)^2\color{red}{k^2} + (k+1)^2 \color{red}{4(k+1)}\\
&= (k+1)^2\color{red}{k^2} + \color{red}{4}(k+1)^3 \\
&=\text{right hand side}
\end{align}\]

Answer 27

you need to know below identity
\[a^2+2ab+b^2 = (a+b)^2\]

Answer 28

\[k^2+4k+4 = k^2+2\cdot k\cdot 2 + 2^2 = (k+2)^2\]

Answer 29

agh