SOLUTION: can someone help me write to simplest form 4r^2-25s^2 over 2r^2+3rs-20s^2

Question 75935: can someone help me write to simplest form
4r^2-25s^2
over
2r^2+3rs-20s^2
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
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The numerator of this fraction can easily be factored if you recognize that is is the difference
of two squared terms. The form is:
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This says that if you have the difference of two squares you can factor it into the product
of the sum and difference of their square roots.
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Applying this rule to the numerator of the fraction we can see that the square root of
each of the terms in the numerator are 2r and 5s. So the factored form of the numerator
is and the problem now takes the form:
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The following is a schoolwork suggestion. This problem was set up to teach you something.
It is likely that the lesson is to learn to cancel a factor in the denominator with a like
factor appearing in the numerator. Therefore, it likely that the denominator also contains
one of the factors that we found in the numerator. That's a clue to what we should look
for when we factor the denominator. So let's factor the denominator next ...
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We can tell from the first term in the denominator that it factors into
2r and r. So we know that our factored denominator is of the form:
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(2r ______ ) * (r ______ )
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The last term in the denominator tells us something. Because it has a minus
sign, one of its factors must be positive and one must be negative. (If they both were
positive or both negative they would multiply together to give a positive, not a negative
term.) We also know that both the factors contain an s, so we can write the factored
form of the denominator as:
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(2r ____ s)*(r ____ s)
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All we need to do is find the positions of the + and - signs and the numbers for the blanks.
We can now guess that 5 is the number for the blank in the factor containing 2r because
that would make it possible to cancel that term with a like term in the numerator.
So we can guess that the factor is now of the form:
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(2r ___ 5s)*(r ___ s)
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Now we can tell that the number in the second factor must be 4. Why? Because the 5s at
the end of the first factor must multiply it and result in . So we can now
write our factored form of the denominator as:
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(2r ___ 5s)*(r ___ 4s)
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All we have to do now is to find the position of the + and - signs that go into the blanks.
We know that 2r times 4s and then r times 5s produce two products that add together
to give + 3rs of the original denominator. These products are 8rs and 5rs. For these terms
to combine to give +3s, we can see that 8rs must be positive and 5rs must be negative.
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For the 8rs to be positive, the 4s must be positive. And therefore the 5s must be negative.
This means that the factors in the denominator are:
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(2r - 5s)*(r + 4s)
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[You can always check this by multiplying the two terms together to see if the result
really is the original form of the denominator in the problem.]
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Now substitute this pair of factors for the original denominator and the problem is then:
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Next cancel the like terms in the denominator and numerator:
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And you are left with the answer:
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Hope this helps you to see how to simplify the original problem into its "reduced"
form.
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