Last one: Simplify completely quantity x squared minus 3 x minus 54 over quantity x squared minus 18 x plus 81 times quantity x squared plus 12 x plus 36 over quantity x plus 6

Question

campbell_st · Answer

well factor all the quadratics... 

1. then multiply numerators and denominators

2. when cancel common factors... 

steps 1 and 2 are interchangeable

j2lie · Answer

x^2-3x-54/x^2-18x+81*x^2+12x+36/6

anonymous · Answer

Yes, that's it. 

I tried factoring this one @campbell_st but don't know what I'm supposed to do if no numbers are the product of -53 and the sum of -3 in the top left, etc.,

campbell_st · Answer

ok... so start bit by bit

factor 

$$x^2 -3x -54 $$

campbell_st · Answer

then factor 

$$x^2 -18x + 81$$

campbell_st · Answer

then factor 

$$x^2 + 12x + 36$$

j2lie · Answer

you factor it with the Magic X.

anonymous · Answer

Ok, well I know the left side denominator is (x - 9) ( x + 9)...

campbell_st · Answer

nope... not quite right

campbell_st · Answer

left denominator is a trinomial

you factorisation gives  x^2 - 81

campbell_st · Answer

I'll do the right side denominator 

(x + 6)

anonymous · Answer

Ok, right since they cancel out, just saw that! 
The right top is (x + 6) (x + 6)? 
Still can't see how the top left is facotorable.

campbell_st · Answer

ok... so the problem looks like 

|dw:1391896674904:dw|

campbell_st · Answer

well what are the factors of -54... that add to -3... the larger factor is negative... ths smaller is positive

anonymous · Answer

I really don't see any numbers that do, though I'm probably wrong!
And the bottom left is x^2 = 81

anonymous · Answer

Wait! Somewhere along the lines of 6 and 9!

campbell_st · Answer

the left denominator is a perfect square

(x + a) = x^2 + 2ax + a^2

campbell_st · Answer

well thats a start... 

which is negative so that 

9 and 6 give add to -3

anonymous · Answer

-9 and 6?

anonymous · Answer

So (x - 9) and (x - 3) ?

campbell_st · Answer

great so its 

|dw:1391897040082:dw|

and with the perfect square is 

(x -a) = x^2 - 2ax + a^2

anonymous · Answer

So the 6's all cancel out? Do I have to put the x^2 - 81 into that form?

campbell_st · Answer

no its not x^2 - 81

its a perfect square... it has 3 terms... 

it will be

(x -a)(x-a) = x^2 -2ax + a^2

campbell_st · Answer

andfinish factoring before cancelling

anonymous · Answer

I'm sorry, I don't know how to put it into that form.:/

campbell_st · Answer

$$x^2 -18x + 81 = (x -9)(x -9)$$

so you problem is now 

|dw:1391897330259:dw|

now cancel the common factors... what's left..?

anonymous · Answer

(x + 6) / (x - 9) ?

anonymous · Answer

Actually (x + 6)^2 / (x - 9) ?

campbell_st · Answer

|dw:1391897503370:dw|

now judt distribute the numerator for the final asnswer