math question on simplest form?

Question

these are the problems.......
#1) combine and express in simplest form:

y - 20 2
-------------- + --------
y^2 - 16 y - 4

#2) expressed in simplest form,

x^2 = x - 6 x + 2
------------------ * -------------- , x does not = 2, o, -2,is equivalent to?
x^2 - 4 x + 3



my question is for the multiplying do you factor and simplify and for adding you factor then combine?? how can u tell when and wat to do?

Answer 1

You need to factor them in any case, whether for simplifying or for combining, so start off by doing that.

(y-20) / (y^2 - 16) + 2 / (y-4)
= (y-20) / ((y-4)(y+4)) + 2 / (y-4)

Now check to see if there's anything we can do to simplify the individual fractions. Here there isn't, because y-20 isn't a factor of the denominator on the left and 2 isn't a factor of the denominator on the right. So now we find the lowest common denominator. Since (y-4) is a factor of the denominator on the left, the lcd is clearly (y-4)(y+4). So we get

= (y-20) / ((y-4)(y+4)) + 2(y+4) / ((y-4)(y+4))

Now combine, simplify and factorise the numerator:

= (y-20 + 2y+8) / ((y-4)(y+4))
= (3y-12) / ((y-4)(y+4))
= 3(y-4) / ((y-4)(y+4))

Now cancel out any common factors:
= 3 / (y+4).

For #2 I think this is the problem you're trying to do; it's hard to tell because the spacing is completely screwed up... and also I think you've typed = when you meant + (or possibly -...)
[(x^2 + x - 6) / (x^2 - 4)] × [(x + 2) / (x + 3)]

Again, the first thing to do is factorise:
= [((x+3)(x-2)) / ((x-2)(x+2))] × [(x+2) / (x+3)]

Cancel common factors within each fraction:
= [(x+3) / (x+2)] × [(x+2) / (x+3)]

Now multiply and cancel additional common factors:
= [(x+3) (x+2)] / [(x+2) (x+3)]
= 1.

With practice you can combine the last few steps, and cancel within the same fraction and across fractions at the same time (but only when multiplying, never when adding!).