Question

How do you simplify `\frac { x ^ { 2} - x - 72} { x ^ { 2} + 13x + 40} \div \frac { x ^ { 2} + 5x + 6} { x ^ { 2} + 7x + 10}`?

Answer 1

`(x-9)/(x+3)`

Explanation:

`color(blue)("Initial thoughts")`

We need to experiment with factorizing to see if there is anything we can cancel out.

`x^2+5x+6" "->" "2xx3 = 6 and 2+3=5`

`color(white)("ddd.dddddddd")->(x+2)(x+3)`
....................................................................................

`x^2+7x+10" "->" "2xx5=10 and 2+5=7`

`color(white)("dddddddddddd")->(x+2)(x+5)`
....................................................................................

`x^2+13x+40" "->" "5xx8=40 and 5+8=13`

`color(white)("ddddddddddddd")->(x+5)(x+8)`
....................................................................................

`x^2-x-72" "->" "9xx8=72 and 8-9=-1`

`color(white)("dddd.ddddddd")->(x-9)(x+8)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together")`

Invert and multiply the divisor:

`((x-9)cancel((x+8)))/(cancel((x+5))cancel((x+8))) xx(cancel((x+2))cancel((x+5)))/(cancel((x+2))(x+3))`

`(x-9)/(x+3)`

Answer 2

`(x-9)/(x+3)`

Explanation:

`(x^2 - x - 72)/(x^2 + 13x + 40) div (x^2 + 5x + 6)/(x^2 + 7x + 10)`

Lets start by rewriting the problem such that all of the terms are combined into a single numerator and denominator. We can eliminate the division symbol by inverting the second term and switching to multiplication.

`(x^2 - x - 72)/(x^2 + 13x + 40) xx (x^2 + 7x + 10)/(x^2 + 5x + 6)`

Now we can combine both terms into a single fraction.

`((x^2 - x - 72)(x^2 + 7x + 10))/((x^2 + 13x + 40)(x^2 + 5x + 6))`

We have four quadratic statements separated by parenthesis. By factoring each we may be able to find terms that cancel each other out.

`((x+8)(x-9)(x+5)(x+2))/((x+8)(x+5)(x+2)(x+3))`

Sure enough, we see that `x+8`, `x+5`, and `x+2` are all present in both the numerator and denominator.

`(color(red)cancel(color(black)((x+8)))(x-9)color(red)cancel(color(black)((x+5)))color(red)cancel(color(black)((x+2))))/(color(red)cancel(color(black)((x+8)))color(red)cancel(color(black)((x+5)))color(red)cancel(color(black)((x+2)))(x+3))`

After eliminating the like terms, we are left with the reduced solution to the problem.

`(x-9)/(x+3)`