SOLUTION: Simplify and reduce to lowest terms 1/(√t-5)-3/(√t+5)

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Question 235274: Simplify and reduce to lowest terms
1/(√t-5)-3/(√t+5)
Found 2 solutions by stanbon, MathPro:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Simplify and reduce to lowest terms
1/(√t-5)-3/(√t+5)
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gcd = sqrt(t-5)*sqrt(t+5) = sqrt(t^2-5)
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Rewrite each fraction with the gcd as its denominator:
(sqrt(t+5))/gcd - 3(sqrt(t-5))/gcd
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Combine the numerators over the gcd:
[sqrt(t+5) - 3sqrt(t-5)]/[sqrt(t^2-5)]
================================
Cheers,
Stan H.
Answer by MathPro(14)   (Show Source): You can put this solution on YOUR website!
Good evening,
First I’ll start with some notation. I’ll write the square root in the following way:
t^1/2
The problem:
1/(t^1/2 – 5) - 3(t^1/2 + 5)=0
Let’s digress, if you are asked to add two fractions the denominator must be the same. For example:
1/2 + 1/3 = 3/6 + 2/6 = 5/6
So let’s apply this to our problem and multiply the second term by (t^1/2 -5) / (t^1/2 -5) to get:
1/(t^1/2-5) -3[(t^1/2 -5)(t^1/2 + 5)] / (t^1/2 – 5) = 0
Simplify by multiplying both sides of the equation by the denominator to get:
1 -3[(t^1/2 -5)(t^1/2 + 5)]=0
Multiply out the 2nd term:
1 -3[t^1/2 x t^1/2 + 5t^1/2 - 5t^1/2 – 25]= 1 -3[t-25]= 1 -3t + 75 =0
So now we have:
1 -3t + 75 =0
t=76/3 = 25.33333333….
Let’s check the result by substituting for t in the equation. If I do that I get:
30.09966 – 30.09966 = 0
The left-hand-side is equal to the right-hand-side, so the answer is correct.
Good Luck!
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