Given the equation:
$$\frac{1}{\left(x - 3\right)^{2}} = 4 + \frac{3}{x - 3}$$
Multiply the equation sides by the denominators:
(-3 + x)^2
we get:
$$\frac{\left(x - 3\right)^{2}}{\left(x - 3\right)^{2}} = \left(4 + \frac{3}{x - 3}\right) \left(x - 3\right)^{2}$$
$$1 = \left(x - 3\right) \left(4 x - 9\right)$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$1 = \left(x - 3\right) \left(4 x - 9\right)$$
to
$$- 4 x^{2} + 21 x - 26 = 0$$
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
$$a = -4$$
$$b = 21$$
$$c = -26$$
, then
D = b^2 - 4 * a * c =
(21)^2 - 4 * (-4) * (-26) = 25
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2$$
$$x_{2} = \frac{13}{4}$$