# Alg. 2 question

If A=9^9999 + 9^-9999 and B= 9^9999 - 9^-9999.
Then find the value of A^2 -B^2 + 1.

would i just take the the numbers and square them to get the value. What should i do.

$$A^2 - B^2 = ( A+B )( A-B )$$

for any $A, B$
That should simplify it a great deal

so would the values be 0 or 18. this question is confusing to me even when you did simplify it for me

James R
Homework Helper
Gold Member
Put p = 9^9999 and q= 9^-9999

Then A = p+q and B = p-q

You want:

$$A^2 + B^2 - 1 = (p+q)^2 + (p-q)^2 - 1 = (p^2 + q^2 + 2pq) + (p^2 +q^2 - 2pq) - 1$$
$$= 2p^2 + 2q^2 - 1 = 2(9^{9999}) + 2(9^{-9999}) - 1$$

Curious3141
Homework Helper
James R said:
Put p = 9^9999 and q= 9^-9999

Then A = p+q and B = p-q

You want:

$$A^2 + B^2 - 1 = (p+q)^2 + (p-q)^2 - 1 = (p^2 + q^2 + 2pq) + (p^2 +q^2 - 2pq) - 1$$
$$= 2p^2 + 2q^2 - 1 = 2(9^{9999}) + 2(9^{-9999}) - 1$$
Unnecessarily complicated, and I'm afraid you read the question wrongly.

$$A = 9^{9999} + 9^{-9999}$$ and $$B = 9^{9999} - 9^{-9999}$$

$$A + B = (2)(9^{9999})$$ and $$A - B = (2)(9^{-9999})$$

$$A^2 - B^2 + 1 = (A + B)(A - B) + 1 = (4)(9^{9999})(9^{-9999}) + 1 = 4 + 1 = 5$$

James R
Homework Helper
Gold Member
Curious3141 said:
Unnecessarily complicated, and I'm afraid you read the question wrongly.
I would argue that my solution is no more complicated than yours. You are, however, correct that I copied the question wrongly. My correct solution is:

$$A^2 - B^2 + 1 = (p+q)^2 - (p-q)^2 + 1 = (p^2 + q^2 + 2pq) - (p^2 +q^2 - 2pq) + 1$$
$$=4pq + 1 = 4(9^{9999})(9^{-9999}) + 1 = 4 + 1 = 5$$

So, we agree.

they both seem right to me just that the letters are changed and makes it a little organized and not messy. So the right answer is 4+1 = 5.
Thanks for the help.

I was wondering if you an explantation for what you did since my teachers requires how we got it instead of just stating a thoure (how ever you spell it) about how we got it. it is kind of stuiped but makes perfect sense to him.

Curious3141
Homework Helper
James R said:
I would argue that my solution is no more complicated than yours. You are, however, correct that I copied the question wrongly. My correct solution is:

$$A^2 - B^2 + 1 = (p+q)^2 - (p-q)^2 + 1 = (p^2 + q^2 + 2pq) - (p^2 +q^2 - 2pq) + 1$$
$$=4pq + 1 = 4(9^{9999})(9^{-9999}) + 1 = 4 + 1 = 5$$

So, we agree.
I feel it's needlessly complicated to expand out the square terms, you can factorise immediately without doing that. With your notation, it would simply be :

$$(p + q)^2 - (p - q)^2 + 1 = (p + q + p - q)(p + q - p + q) + 1 = (2p)(2q) + 1 = 4pq + 1$$

and I personally think that is simpler, IMHO. But let's not split hairs.

James R