Answer:
steps below
Step-by-step 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)
Step-by-step explanation:
We first need to simplify the quadratic numerators and denominators. We'll use the fact that any equation of form
[tex]ax^2 + bx + c = 0[/tex]
where it has solutions [tex]x_1[/tex] and [tex]x_2[/tex] can be written as
[tex]a(x-x_1)(x-x_2)[/tex]
Where [tex]b = -a(x_1 + x_2)[/tex] and [tex]c = ax_1x_2[/tex]
Fortunately, here a = 1 for all equations, so let's learn how to solve for a=1 because it's a bit simpler.
[tex]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)\\~\\~\\\frac{x^2-2x-8}{x^2-3x-10} = \frac{(x-4)(x+2)}{(x-5)(x+2)} = \frac{x - 4}{x-5}[/tex]
See, we've simplified the first term of the subtraction. Now let's do the same with the right side:
[tex]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)\\~\\~\\\frac{x^2 - 2x - 3}{x^2 + 7x + 6} = \frac{(x-3)(x+1)}{(x+1)(x+6)} = \frac{x-3}{x+6}[/tex]
See, we've simplified the second term. Now we need to bring them to the common denominator:
[tex]\frac{x^2-2x-8}{x^2-3x-10} - \frac{x^2-2x-3}{x^2+7x+6} = \frac{x-4}{x-5} - \frac{x-3}{x+6} = \frac{(x-4)(x+6)}{(x-5)(x+6)} - \frac{(x-3)(x-5)}{(x+6)(x-5)} =\\~\\= \frac{(x-4)(x+6) - (x-3)(x-5)}{(x+6)(x-5)} = \frac{(x^2 + 2x - 24) - (x^2 - 8x + 15)}{(x+6)(x-5)} =\\~\\= \frac{10x-39}{(x+6)(x-5)}[/tex]
We could simplify even further if we can find some a and b so that
[tex]10x - 39 = a(x+6) + b(x-5)[/tex]
then
[tex]\frac{10x-39}{(x+6)(x-5)} = \frac{a}{x-5} + \frac{b}{x+6}[/tex]
So let's try!
[tex]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\\~\\\frac{10x-39}{(x+6)(x-5)} = \frac{1}{x-5} + \frac{9}{x + 6}[/tex]
And that's the simplest form.
