Do the operations and simplify:?

Question

a + 6 4a - 20
_____ * ______
25 - a^2 4a + 24

Answer 1

So you want to multiply:

[(a + 6) / (25 - a^2)] x [(4a - 20) / (4a + 24)]

Your first step is to factor everything. Note that 25 - a^2 factors as a difference of squares, and other things have a factor that can be pulled out of them. Let's factor *everything*.

[(a + 6) / {(5 - a)(5 + a)}] x [ {4(a - 5)} / {4(a + 6)} ]

Recall that when mutiplying fractions together, they cancel with each other not only vertically (common terms in numerator and denominator), but also diagonally. The (a + 6) in the first fraction cancels with the (a + 6) in the second fraction. The two 4s in the second fraction cancel each other out.

[1 / {(5 - a)(5 + a)}] x [(a - 5) / 1 ]

Another thing to notice is that the factors that can potentially cancel each other out, cannot at the moment, because (a - 5) and (5 - a) are not quite the same. The technique you have to use is known as the "negative one technique". This basically states that whenevery ou have a difference of terms (x - y), you can reverse them by factoring out -1. That is, (x - y) is the same as (-1)(y - x). This is what we do.

[1 / {(5 - a)(5 + a)}] x [(a - 5) / 1]

Applying the -1 technique to the second fraction, we get

[1 / {(5 - a)(5 + a)}] x [(-1)(5 - a) / 1 ]

And now we can cross cancel the (5 - a), to get

[1 / (5 + a)] x [-1 / 1]

This simplifies into

[1/(5+a)] [-1], or

-1/(5+a)

Answer 2

Well, first I'll reformat it a little:

{(a + 6) / (25 - a^2)} * {(4a - 20) / (4a + 24)} and then we factor it:

{ (a + 6) / [(5 - a) * (5 + a)] } * { [4 * (a - 5)] / [4 * (a + 6)]}.

You can see without too much trouble the 4's cancel (in the last half). I'll drop them as cancelled and move the terms around so we have the dividends together and the divisors together. I will also factor (5 - a) to be: –1 * (a - 5) and change the (5 + a) around to (a + 5):

{ (a + 6) * (a - 5) } / { –1 * (a - 5) * (a + 5) * (a + 6) }

Now it should be pretty clear to see the (a + 6)'s and (a - 5)'s cancelling and dividing by –1 is the same as multiplying by –1 so:

–1 / (a + 5) is the final reduced form. Check it by substituting some value, 2 say, into the starting form and the final form and verify the results are the same (2 yields –1/7).

Finally, since the initial form had division by (25 - a^2) and (4a + 24), a cannot equal 5, –5 or –6.

Answer 3

(a+6/25-a^2) x (4a-20/4a+24)
= (a+6)(4a-20) / (25-a^2)(4a+24)
= 4a^2-20a+24a-120 / 100a+600-4a^3-24a^2
=4a(a-5+6)-120 / 4a(25-a^2-6a)+600
=a+1-120 / 25-a^2-6a +600
=a-119 / -a^2-6a+625

Answer 4

a + 6 4a - 20
-----------* -----------
25 - a^2 4a + 24

a + 6 4(a - 5)
------------------*---------
-(a - 5)(a + 5) 4(a + 6)

1 * 4
-------
-(a + 5)(4)

-(1/(a + 5))

Answer 5

Please ignore all the ...... it's the only way I could get the spacing right.


a+6.....................4(a-5)
----------------* ---------
(5-a)(5+a)...........4(a+6)


Cancel the (a+6) and rewrite the first denominator:

1.......................4(a-5)
------------------- * ----------
(-1)(-5+a)(a+5) ........ 4


Cancel the 4 and again rewrite the first denominator:
1.......................(a-5)
------------------- * ---------
(-1)(a-5)(a+5)...........1

Now cancel the (a-5):
1................(1)
------------ * ---------
(-1)(a+5)..............1

The second fraction simplifies to 1, and that means your answer is:

-1
-----
a+5