Solve for x without using the quadratic formula

Question

Some context: I'm doing an inverse transformation method where I have the probability density function split in three parts. The first part is:

$$ f_1:\frac{x-6}{8} $$

For $ 6 < x < 8 $.

I know by integrating $ f_1 $ that the range goes from $ 0 $ to $ 0.25 $.

So now I'm trying solve for $ x$.

$$ u=\int^x_6\frac{x-6}{8}dx $$

Where $ u $ is in the range $ [0, 0.25] $ (and is taken as a random variable with uniform distribution).

I remember that the professor was able to solve for $ x $ without doing quadratic formula. Because I have an exam tomorrow and it will take long to solve (and it will be a lot harder if I have a greater degree polynomial), I need to remember. Anyone help?

Answer 1

Will this work?

Set

$y = x - 6; \tag{1}$

then

$u = \int_6^x \dfrac{x - 6}{8} dx = \int_0^y \dfrac{y}{8}dy$ $ = \dfrac{1}{8}\int_0^y ydy = \dfrac{y^2}{16}; \tag{2}$

thus

$y^2 = 16u, \tag{3}$

or

$y = 4\sqrt{u}. \tag{4}$

Not quite the quadratic formula!

Taking it one step further,

$x = y + 6 = 4\sqrt{u} + 6. \tag{5}$