Question

Simplify the expression (Image below) Precalculus please help! D:

Answer 1

`\boxed{\sf (x-4)/(x^2-2x) }`

Explanation:

`\sf \cfrac{x+6}{x^2+3x}-\cfrac{5}{x^2+x-6}`

Factor x² + 3x, and x²+ x - 6

`\sf \cfrac{x+6}{x(x+3)} -\cfrac{5}{(x-2)(x+3}`

To add or subtract expressions, we expand them to make their denominators the same, LCM of x(x+3) and (x-2)(x+3) is x(x-2) (x+3).

Multiply
`\sf (x+6)/(x(x+3))* \sf (x-3)/(x-2)` and
`(5)/((x-2)(x+3)) * (x)/(x)`

`\sf \cfrac{(x+6)(x-2)}{x(x-2)(x+3)}-\cfrac{5x}{x(x-2)(x+3)}`

Here,
`\sf ((x+6)(x-2))/(x(x-2)(x+3))` and
`\sf (5x)/(x(x-2)(x+3)` have the same denominators, we will subtract them by subtracting their numerators:

`\sf \cfrac{(x+6)(x-2)-5x}{x(x-2)(x+3)}`

*Multiply (x+6)(x-2)-5x:

`\sf \cfrac{x^2-2x+6x-12-5x}{x(x-2)(x+3)}`

Combine like terms: x² - 2x + 6x - 12 - 5x

`\sf \cfrac{-x-12+x^2}{x(x-2)(x+3)}`

Now, factor expressions that are not already factored:

`\sf \cfrac{(x-4)(x+3)}{x(x-2)(x+3)}`

`\sf \cfrac{x-4}{x(x-2)}`

Expand:

`\sf \cfrac{x-4}{x^2-2x}`

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