shapes and angle

Given that

2x−1 : x−4  =  16x+1 : 2x−1

find the possible values of x.

(We can see that   x ≠ 4    and    x ≠ 1/2   , because those values cause a zero in the denominator.)

\(\frac{2x-1}{x-4}=\frac{16x+1}{2x-1}\)                                                   Cross multilpy.

\((2x-1)(2x-1)=(16x+1)(x-4) \)

\(4x^2-4x+1=16x^2-63x-4\)                 Subtract  4x²  from both sides of the equation.

\(-4x+1=12x^2-63x-4\)                          Add  4x  to both sides of the equation.

\(1=12x^2-59x-4\)                                       Subtract  1  from both sides of the equation.

\(0=12x^2-59x-5\)                                       Use the quadratic formula to solve for  x  .

\(x = {59 \pm \sqrt{(-59)^2-4(12)(-5)} \over 2(12)} \\~\\ x = {59 \pm 61 \over 24} \\~\\ x=5\qquad\text{and}\qquad x=-\frac1{12}\)

Also..I did work on the question 20 at first if you need help on that one...I had a whole answer ready for it then I saw you changed it! Haha 

2x -1 :  x  - 4  =  16x  + 1 : 2x -1

So....we can write this as

[ 2x - 1 ] / [x - 4 ]  = [ 16x + 1 ] / [2x -1]       cross-multiply

[2x -1 ] [2x -1 ]  = [x - 4] [ 16x + 1 ]   simplify

4x^2  - 4x  + 1   = 16x^2 - 64x  + x - 4

4x^2 - 4x + 1  = 16x^2 -63x - 4       subtract the left side from the right

12x^2  - 59x - 5  = 0       factor

(12x + 1) ( x - 5 )  = 0

Set both factors to 0 and solve.....and x = -1/12      or  x  = 5