Posted by Tammy on Thursday, July 19, 2007 at 10:12pm.
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(3x+4)/(3x-4)+(3x-4)/(3x+4)
--------------------------- Divide top by bottom
(3x-4)/(3x+4)-(3x+4)/(3x-4)
Very Big Problem and help would be greatly appreciated
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Algebra - Reiny, Thursday, July 19, 2007 at 11:59pm
not that tough once you notice that your question has the pattern
(a/b + b/a)÷(b/a - a/b)
where a = 3x+4 and b=3x-4
this easily reduces to (a^2 + b^2)/(b^2 - a^2)
now replace a and b with 3x+4 and 3x-4 and it simplifies to
(9x^2+16)/(-24x)
hmm, the answer is right (it's in my book) but I don't understand how you simply reduce it to (a^2+b^2) (b^2-a^2)
and from there when I tried to replace a and b with 3x+4 and 3x-4 I don't know how to simplify it or get the answer. Could you write down the steps detailed? I have a bunch of problems that I can't solve that are similar, so I think if I understand the concept of this one I will be able to sove all the other ones.
The fact that there was that nice symmetry to the question, makes it easier than it first appears.
I had it as:
(a/b + b/a)÷(b/a - a/b)
look at (a/b + b/a)
the common denominator for this would be ab, so
a/b + b/a = (a^2 + b^2)/ab
similarly for
(b/a - a/b) the common denominator is ab and this reduces to
(b^2 - a^2)/ab
so now we have
(a^2 + b^2)/ab ÷ (b^2 - a^2)/ab
remember when dividing one fraction by another, we multiply by the reciprocal of the second fraction, so
(a^2 + b^2)/ab ÷ (b^2 - a^2)/ab
=(a^2 + b^2)/ab * ab/((b^2 - a^2)
the ab's cancel and that's how I got
(a^2 + b^2)/(b^2 - a^2)
Now replace the original values of a and b and you should be able to do that yourself.
To simplify the expression (3x+4)/(3x-4)+(3x-4)/(3x+4), we can use the concept of adding fractions.
Let's first focus on the numerator. Notice that we have (3x+4) in the first fraction and (3x-4) in the second fraction. We can rewrite the numerator as (3x+4)(3x-4) since the denominators are the same. Expanding this expression gives us 9x^2 - 16, which becomes the numerator of our simplified expression.
Next, let's focus on the denominator. We have (3x-4) in the first fraction and (3x+4) in the second fraction. We can rewrite the denominator as (3x-4)(3x+4) since the numerators are the same. Expanding this expression gives us 9x^2 - 16, which becomes the denominator of our simplified expression.
Putting it all together, we have (9x^2 - 16)/(9x^2 - 16) as the simplified expression.
Now, let's address the second part of the question: how to simplify (9x^2 - 16)/(-24x) when we substitute a and b with 3x+4 and 3x-4, respectively.
Substituting a and b into the expression (a^2 + b^2)/(b^2 - a^2), we get ((3x+4)^2 + (3x-4)^2)/((3x-4)^2 - (3x+4)^2).
Expanding the squares, we have ((9x^2 + 24x + 16) + (9x^2 - 24x + 16))/((9x^2 - 16) - (9x^2 + 24x + 16)).
Simplifying further, we have (18x^2 + 32)/(16x^2 - 16).
Finally, factoring out a common factor of 16 from both terms in the denominator, we get (18x^2 + 32)/(16(x^2 - 1)).
So, the simplified expression when substituting 3x+4 for a and 3x-4 for b is (18x^2 + 32)/(16(x^2 - 1)).
Remember to always carefully simplify expressions by factoring, canceling out common factors, and expanding when needed.