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72:(x-3)-72:(x+3)=2 equation
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The solution
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[src]
72 72 ----- - ----- = 2 x - 3 x + 3
$$- \frac{72}{x + 3} + \frac{72}{x - 3} = 2$$
Detail solution
Given the equation:
$$- \frac{72}{x + 3} + \frac{72}{x - 3} = 2$$
Multiply the equation sides by the denominators:
-3 + x and 3 + x
we get:
$$\left(x - 3\right) \left(- \frac{72}{x + 3} + \frac{72}{x - 3}\right) = 2 x - 6$$
$$\frac{432}{x + 3} = 2 x - 6$$
$$\frac{432}{x + 3} \left(x + 3\right) = \left(x + 3\right) \left(2 x - 6\right)$$
$$432 = 2 x^{2} - 18$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$432 = 2 x^{2} - 18$$
to
$$450 - 2 x^{2} = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = 450$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -15$$
$$x_{2} = 15$$
$$- \frac{72}{x + 3} + \frac{72}{x - 3} = 2$$
Multiply the equation sides by the denominators:
-3 + x and 3 + x
we get:
$$\left(x - 3\right) \left(- \frac{72}{x + 3} + \frac{72}{x - 3}\right) = 2 x - 6$$
$$\frac{432}{x + 3} = 2 x - 6$$
$$\frac{432}{x + 3} \left(x + 3\right) = \left(x + 3\right) \left(2 x - 6\right)$$
$$432 = 2 x^{2} - 18$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$432 = 2 x^{2} - 18$$
to
$$450 - 2 x^{2} = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = 450$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2) * (450) = 3600
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -15$$
$$x_{2} = 15$$
Sum and product of roots
[src]
sum
-15 + 15
$$-15 + 15$$
=
0
$$0$$
product
-15*15
$$- 225$$
=
-225
$$-225$$
-225