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(12)/(x-1)-(8)/(x+1)=2 equation
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The solution
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[src]
12 8 ----- - ----- = 2 x - 1 x + 1
$$- \frac{8}{x + 1} + \frac{12}{x - 1} = 2$$
Detail solution
Given the equation:
$$- \frac{8}{x + 1} + \frac{12}{x - 1} = 2$$
Multiply the equation sides by the denominators:
-1 + x and 1 + x
we get:
$$\left(x - 1\right) \left(- \frac{8}{x + 1} + \frac{12}{x - 1}\right) = 2 x - 2$$
$$\frac{4 \left(x + 5\right)}{x + 1} = 2 x - 2$$
$$\frac{4 \left(x + 5\right)}{x + 1} \left(x + 1\right) = \left(x + 1\right) \left(2 x - 2\right)$$
$$4 x + 20 = 2 x^{2} - 2$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$4 x + 20 = 2 x^{2} - 2$$
to
$$- 2 x^{2} + 4 x + 22 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 4$$
$$c = 22$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 1 - 2 \sqrt{3}$$
$$x_{2} = 1 + 2 \sqrt{3}$$
$$- \frac{8}{x + 1} + \frac{12}{x - 1} = 2$$
Multiply the equation sides by the denominators:
-1 + x and 1 + x
we get:
$$\left(x - 1\right) \left(- \frac{8}{x + 1} + \frac{12}{x - 1}\right) = 2 x - 2$$
$$\frac{4 \left(x + 5\right)}{x + 1} = 2 x - 2$$
$$\frac{4 \left(x + 5\right)}{x + 1} \left(x + 1\right) = \left(x + 1\right) \left(2 x - 2\right)$$
$$4 x + 20 = 2 x^{2} - 2$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$4 x + 20 = 2 x^{2} - 2$$
to
$$- 2 x^{2} + 4 x + 22 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 4$$
$$c = 22$$
, then
D = b^2 - 4 * a * c =
(4)^2 - 4 * (-2) * (22) = 192
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 1 - 2 \sqrt{3}$$
$$x_{2} = 1 + 2 \sqrt{3}$$
Rapid solution
[src]
___ x1 = 1 - 2*\/ 3
$$x_{1} = 1 - 2 \sqrt{3}$$
___ x2 = 1 + 2*\/ 3
$$x_{2} = 1 + 2 \sqrt{3}$$
x2 = 1 + 2*sqrt(3)
Sum and product of roots
[src]
sum
___ ___ 1 - 2*\/ 3 + 1 + 2*\/ 3
$$\left(1 - 2 \sqrt{3}\right) + \left(1 + 2 \sqrt{3}\right)$$
=
2
$$2$$
product
/ ___\ / ___\ \1 - 2*\/ 3 /*\1 + 2*\/ 3 /
$$\left(1 - 2 \sqrt{3}\right) \left(1 + 2 \sqrt{3}\right)$$
=
-11
$$-11$$
-11