I really need your help in solving this problem. I appreciate your time and efforts. Have a terrific day :) Here is the question:
For each expression, perform the indicated operations and simplify, if possible. Solve each equation and check the result.
Two part question:
A. u^2 +1/u^2-u -u/u-1
B. u^2 +1/u^2-u -u/u-1 = 1/u
I assume you mean
(u^2+1)/(u^2-u) - u/(u-1)
Clearly the LCD is u(u-1)
(u^2+1 - u(u+1)) / u(u-1)
(-u+1) / u(u-1)
Now put both sides over the LCD, throw out u=0,1 and set the numerators equal.
Then solve for u.
What do you get?
I am not sure but I will write all of this down. Thank you Steve.
Yours,
JAY
I'm sure that by now you have figured out that I made an error:
(u^2+1 - u^2) / u(u-1)
1 / u(u-1)
Now set that equal to 1/u
1 / u(u-1) = 1/u
1 = u-1
u = 2
check:
5/2 - 2/1 = 1/2
Of course! I'll be glad to help you solve the problem. Let's break it down step by step.
Part A:
The given expression is: u^2 + 1/u^2 - u - u/(u - 1)
To simplify this expression, we'll start by finding a common denominator for the last two terms.
For the term -u/(u - 1), we can multiply the numerator and denominator by (u + 1) to get a common denominator of (u - 1)(u + 1):
-u/(u - 1) = -u(u + 1)/[(u - 1)(u + 1)]
Now we can combine the terms that have the same denominator:
u^2 + 1/u^2 - u - [u(u + 1)/[(u - 1)(u + 1)]]
Next, we need to find a common denominator for the first two terms. The common denominator will be u^2:
u^2 + 1/u^2 - u(u^2)/[u^2(u - 1)(u + 1)] - [u(u + 1)/[(u - 1)(u + 1)]]
Now, we can combine all the terms over the common denominator:
[u^4 + 1 - u^3(u - 1) - u(u + 1)]/[u^2(u - 1)(u + 1)]
Expanding the numerator:
[u^4 + 1 - u^4 + u^3 - u^2]/[u^2(u - 1)(u + 1)]
Canceling out the like terms:
[u^3 - u^2 + 1]/[u^2(u - 1)(u + 1)]
Now we have the simplified expression.
Part B:
Given: u^2 + 1/u^2 - u - u/(u - 1) = 1/u
We can substitute the simplified expression from Part A into the equation:
[u^3 - u^2 + 1]/[u^2(u - 1)(u + 1)] = 1/u
To solve this equation, we'll multiply both sides of the equation by the common denominator, which is u * [u^2(u - 1)(u + 1)]:
[u^3 - u^2 + 1] = [u^2(u - 1)(u + 1)]
Now, we'll expand the equation and bring all terms to one side:
u^3 - u^2 + 1 - (u^3 - u^2(u - 1)(u + 1)) = 0
Simplifying further:
u^3 - u^2 + 1 - (u^3 - (u^4 - u^2)(u + 1)) = 0
Expanding the equation:
u^3 - u^2 + 1 - u^3 + u^4(u + 1) - u^2(u + 1) = 0
Combining like terms:
u^4(u + 1) - u^2(u + 1) - u^2 + 1 = 0
Now, we'll collect like terms:
(u^4 - u^2)(u + 1) - (u^2 - 1) = 0
Factoring out common terms:
u^2(u^2 - 1) - (u^2 - 1) = 0
(u^2 - 1)(u^2 - 1) = 0
Now, we have the equation (u^2 - 1)^2 = 0
To solve this equation, we'll apply the zero-product property:
u^2 - 1 = 0
Adding 1 to both sides:
u^2 = 1
Taking the square root of both sides (considering both positive and negative square roots):
u = ±1
So, the solutions for the equation are u = 1 and u = -1.
To check the solutions, substitute them back into the original equation.
When u = 1:
1^2 + 1/1^2 - 1 - 1/(1 - 1) = 1/1
1 + 1 - 1 - undefined = 1
2 - undefined = 1
The equation holds true.
Similarly, when u = -1:
(-1)^2 + 1/(-1)^2 - (-1) - (-1)/(-1 - 1) = 1/(-1)
1 + 1 - (-1) - (-1)/(-2) = -1
2 + 1 + 1/2 = -1
2.5 ≠ -1
The equation does not hold true for u = -1.
Therefore, the only valid solution is u = 1.
I hope this explanation clarifies the steps involved in solving the problem. Feel free to ask if you have any further questions!