Question

Simplify this problem
`(9x)/(4x - 4) + \frac{ {x}^(2) + 6x }{ {x}^(2) + 5x - 6}`

Answer 1

`(9x)/(4x-4)+(x^2-6x)/(x^2+5x-6)`

1. Factor the denominators as follow:

`\begin{gathered} 4x-4=4(x-1) \\ \\ \\ x^2+5x-6 \\ =x^2+6x-x-6 \\ =x(x+6)-(x+6) \\ =(x-1)(x+6) \\ \\ \\ \\ \\ (9x)/(4(x-1))+(x^2-6x)/((x-1)(x+6)) \end{gathered}`

2. Write the expresion with the less common denominator:

Multiply the first fraction by (x+6) (both parts, numerator and denominator):

`(9x)/(4(x-1))\cdot(x+6)/(x+6)=(9x(x+6))/(4(x-1)(x+6))`

Multiply the second fraction by 4 (both parts, numerator and denominator):

`(x^2-6x)/((x-1)(x+6))\cdot(4)/(4)=(4(x^2-6x))/(4(x-1)(x+6))`

Rewrite the expression with the less common denominator:

`\begin{gathered} (9x(x+6))/(4(x-1)(x+6))+(4(x^2-6x))/(4(x-1)(x+6)) \\ \\ =(9x(x+6)+4(x^2-6x))/(4(x-1)(x+6)) \end{gathered}`

3. Remove parentheses and simplify:

`\begin{gathered} \frac{9x^2^{}+54x+4x^2-24x}{(4x-4)(x+6)} \\ \\ =(13x^2+30x)/(4x^2+24x-4x-24) \\ \\ =(13x^2+30x)/(4x^2+20x-24) \end{gathered}`

Then, the given expression simplified is:

`(9x)/(4x-4)+(x^2-6x)/(x^2+5x-6)=(13x^2+30x)/(4x^2+20x-24)`