x-7/x=6 equation

Given the equation:
$$x - \frac{7}{x} = 6$$
Multiply the equation sides by the denominators:
and x
we get:
$$x \left(x - \frac{7}{x}\right) = 6 x$$
$$x^{2} - 7 = 6 x$$
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$x^{2} - 7 = 6 x$$
to
$$x^{2} - 6 x - 7 = 0$$
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
$$a = 1$$
$$b = -6$$
$$c = -7$$
, then

D = b^2 - 4 * a * c = 

(-6)^2 - 4 * (1) * (-7) = 64

Because D > 0, then the equation has two roots.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = 7$$
$$x_{2} = -1$$