Question

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. quantity x plus 6 over quantity x minus 9 the quantity x plus 6 end quantity over the quantity x minus 9 end quantity squared. the quantity x plus 6 end quantity squared over the quantity x minus 9 the quantity x plus 6 end quantity squared over the quantity x minus 9 end quantity squared.

Answer 1

hello


x² - 3 x - 54 = x² + 6 x - 9 x - 54 = x ( x + 6 ) - 9 ( x + 6 ) = ( x + 6 ) ( x - 9 )x² - 18 x + 81 = ( x - 9 )²x² + 12 x + 36 = ( x + 6 )²... = ( x + 6 ) · ( x - 9 ) / ( x - 9 )² * ( x + 6 )² / ( x + 6 ) = ( after cancellation )= ( x + 6 )² / ( x - 9 )

Have a nice day

Answer 2

`(x-6)/(x-9)`

Explanation:

The given expression is

`x^(2) -3x-54 / x^(2) -18x+81`

To solve this division, we could find the solution factors of each expression.

So, let's work on the numerator first

`x^(2) -3x-54`

First, we need to express this in two factors, where the sign of the first factor would be the same as the second term of the given expression (negative). And the sign of the second factor would be the product of the sign of the second factor term and the third term. As follows

`x^(2) -3x-54=(x-a)(x+b)`

So,
`a` and
`b` would be two number which product is 54 and its subtraction is 3. You would find that 9 and 6 are those number because

`9 * 6 = 54\\9 - 6 =3`

Therefore, the solution factors are

`x^(2) -3x-54=(x-9)(x+6)`

Then, we do the same process with the denominator

`x^(2) -18x+81=(x-9)(x-9)`

In this case, we needed to find to number which product is 81 and which sum is 18.

Now, we replace all these factor into the given fraction.

`x^(2) -3x-54 / x^(2) -18x+81=((x-9)(x+6))/((x-9)(x-9)) =(x-6)/(x-9)`

Therefore, most simple form of the given expression is

`(x-6)/(x-9)`