SOLUTION: 1-2/4t-1=8/(4t-1)^2 It says to type the answer as an integer or simplified fraction

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Question 349209: 1-2/4t-1=8/(4t-1)^2
It says to type the answer as an integer or simplified fraction
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If the equation is:
1-2%2F%284t-1%29=8%2F%284t-1%29%5E2
then please put multiple term numerators and denominators in parentheses.

There are several ways to solve this equation. The shorter ways are based on noticing that either 1%2F%284t-1%29 or just 4t-1 are expressions common in the equation. So we can used "clever", temporary substitutions to make the solution easier. Here are 3 solutions, 2 shorter ones and one long, "unclever", straightforward solution:
  1. Use a substitution for 1%2F%284t-1%29. Let q+=+1%2F%284t-1%29. Then your equation becomes:
    1+-+2q+=+8q%5E2
    The equation is now a fairly simple equation to solve (for q):
    0+=+8q%5E2+%2B+2q+-+1
    0+=+%284q+-+1%29%282q+%2B+1%29
    4q - 1 = 0 or 2q + 1 = 0
    q = 1/4 or q = -1/2
    We have solutions for q but we want solutions for t. So we just substitute back in for q:
    1%2F%284t-1%29+=+1%2F4 or 1%2F%284t-1%29+=+-1%2F2
    4t- 1 = 4 or 4t-1 = -2
    4t = 5 or 4t = -1
    t = 5/4 or t = -1/4
  2. Use a substitution for 4t-1. Let q = 4t-1. Substituting this into your equation becomes:
    1+-+2%2Fq+=+8%2Fq%5E2
    To solve this, we'll start by multiplying both sides by q%5E2:
    q%5E2+-2q+=+8
    Subtract 8 from each side:
    q%5E2+-2q+-+8+=+0
    %28q-4%29%28q%2B2%29+=+0
    q - 4 = 0 or q - 2 = 0
    q = 4 or q = -2
    Now we substitute back in for q:
    4t-1 = 4 or 4t-1 = -2
    4t = 5 or 4t = -1
    t = 5/4 or t = -1/4
  3. The long way with no substitutions. We'll start by multiplying both sides by %284t-1%29%5E2 giving:
    %284t-1%29%5E2+-+2%284t-1%29+=+8
    Simplifying we get:
    16t%5E2+-8t+%2B+1+-8t+%2B+2+=+8
    16t%5E2+-16t+%2B+3+=+8
    Subtract 8 from each side:
    16t%5E2+-16t+-+5+=+0
    %284t+-+5%29%284t+%2B+1%29+=+0
    4t - 5 = 0 or 4t + 1 = 0
    4t = 5 or 4t = -1
    t = 5/4 or t = -1/4

Note that all three solutions gave the same results. Keep your eyes open for repeated expressions like there were in this problem. The solutions whichtook advantage of the repeated expressions in your equation made the solution easier. (With a little practice you will eventually be able to do these without actually using a temporary variable.)