Question

Please simplify the difference!: ((x^2-2x-8)/(x^2-3x-10)) - ((x^2-2x-3)/(x^2+7x+6)) Please don't just give me the answer, I actually want to know how to do problems like this--the material the entire course was based on was taught using the learning style I cannot work with whatsoever and I retained none of it. This is more of a study question for me than anything. Please help. This is more of a study question for me than anything. Please help.

Answer 1

Explanation:

We first need to simplify the quadratic numerators and denominators. We'll use the fact that any equation of form

`ax^2 + bx + c = 0`

where it has solutions
`x_1` and
`x_2` can be written as

`a(x-x_1)(x-x_2)`

Where
`b = -a(x_1 + x_2)` and
`c = ax_1x_2`

Fortunately, here a = 1 for all equations, so let's learn how to solve for a=1 because it's a bit simpler.

`x^2 - 2x - 8 = 0\\x^2 - 2x + 1 - 9 = 0\\(x - 1)^2 - 9 = 0\\(x - 1)^2 = 9\\x - 1 = 3 \vee x - 1 = -3\\x = 4 \vee x = -2\\x^2 - 2x - 8 = (x - 4)(x + 2)\\~\\~\\x^2 - 3x - 10 = 0\\x^2 - 3x - 10 = 0\\\\x^2 - 3x + 2.25 - 12.25 = 0\\(x - 1.5)^2 - 12.25 = 0\\(x - 1.5)^2 = 12.25\\x - 1.5 = 3.5 \vee x - 1.5 = -3.5\\x = 5 \vee x = -2\\x^2 - 3x - 10 = (x - 5)(x + 2)\\~\\~\\(x^2-2x-8)/(x^2-3x-10) = ((x-4)(x+2))/((x-5)(x+2)) = (x - 4)/(x-5)`

See, we've simplified the first term of the subtraction. Now let's do the same with the right side:

`x^2 - 2x - 3 = 0\\x^2 - 2x + 1 - 4 = 0\\(x - 1)^2 = 4\\x - 1 = 2 \vee x - 1 = -2\\x = 3 \vee x = -1\\x^2 - 2x - 3 = (x - 3)(x + 1)\\~\\~\\x^2 + 7x + 6 = 0\\x^2 + 7x + 12.25 - 6.25 = 0\\(x + 3.5)^2 = 6.25\\x + 3.5 = 2.5 \vee x + 3.5 = -2.5\\x = -1 \vee x = -6\\x^2 + 7x + 6 = (x + 1)(x + 6)\\~\\~\\(x^2 - 2x - 3)/(x^2 + 7x + 6) = ((x-3)(x+1))/((x+1)(x+6)) = (x-3)/(x+6)`

See, we've simplified the second term. Now we need to bring them to the common denominator:

`(x^2-2x-8)/(x^2-3x-10) - (x^2-2x-3)/(x^2+7x+6) = (x-4)/(x-5) - (x-3)/(x+6) = ((x-4)(x+6))/((x-5)(x+6)) - ((x-3)(x-5))/((x+6)(x-5)) =\\~\\= ((x-4)(x+6) - (x-3)(x-5))/((x+6)(x-5)) = ((x^2 + 2x - 24) - (x^2 - 8x + 15))/((x+6)(x-5)) =\\~\\= (10x-39)/((x+6)(x-5))`

We could simplify even further if we can find some a and b so that

`10x - 39 = a(x+6) + b(x-5)`

then

`(10x-39)/((x+6)(x-5)) = (a)/(x-5) + (b)/(x+6)`

So let's try!

`a + b = 10\\6a - 5b = -39\\b = 10 - a\\6a - 5(10 - a) = -39\\6a - 50 + 5a = -39\\11a = 11\\a = 1\\b = 10 - 1 = 9\\~\\(10x-39)/((x+6)(x-5)) = (1)/(x-5) + (9)/(x + 6)`

And that's the simplest form.

Answer 2

steps below

Explanation:

((x²-2x-8)/(x²-3x-10)) - ((x²-2x-3)/(x²+7x+6))

= ((x+2)(x-4)/(x+2)(x-5)) - ((x+1)(x-3)/(x+1)(x+6))

= (x-4)/(x-5) - (x-3)/(x+6)

= (x-4)(x+6)/(x-5)(x+6) - (x-3)(x-5)/(x-5)(x+6)

= ((x-4)(x+6) - (x-3)(x-5))/(x-5)(x+6)

= ((x²+2x-24) - (x²-8x+15)) / (x-5)(x+6)

= (x²+2x-24-x²+8x-15) / (x-5)(x+6)

= (10x-39) / (x-5)(x+6) ..... (10x-39) / (x²+x-30)