5x(3+4x)-1=54

Solution for 5x(3+4x)-1=54 equation:

Simplifying
5x(3 + 4x) + -1 = 54
(3 * 5x + 4x * 5x) + -1 = 54
(15x + 20x2) + -1 = 54

Reorder the terms:
-1 + 15x + 20x2 = 54

Solving
-1 + 15x + 20x2 = 54

Solving for variable 'x'.

Reorder the terms:
-1 + -54 + 15x + 20x2 = 54 + -54

Combine like terms: -1 + -54 = -55
-55 + 15x + 20x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-55 + 15x + 20x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-11 + 3x + 4x2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-11 + 3x + 4x2)' equal to zero and attempt to solve:

Simplifying
-11 + 3x + 4x2 = 0

Solving
-11 + 3x + 4x2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-2.75 + 0.75x + x2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + 0.75x + 2.75 + x2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + 0.75x + x2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + 0.75x + x2 = 0 + 2.75
0.75x + x2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
0.75x + x2 = 2.75

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 2.75 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 2.75 + 0.140625

Combine like terms: 2.75 + 0.140625 = 2.890625
0.140625 + 0.75x + x2 = 2.890625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 2.890625

Calculate the square root of the right side: 1.700183814

Break this problem into two subproblems by setting 
(x + 0.375) equal to 1.700183814 and -1.700183814.

Subproblem 1
x + 0.375 = 1.700183814

Simplifying
x + 0.375 = 1.700183814

Reorder the terms:
0.375 + x = 1.700183814

Solving
0.375 + x = 1.700183814

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = 1.700183814 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 1.700183814 + -0.375
x = 1.700183814 + -0.375

Combine like terms: 1.700183814 + -0.375 = 1.325183814
x = 1.325183814

Simplifying
x = 1.325183814

Subproblem 2
x + 0.375 = -1.700183814

Simplifying
x + 0.375 = -1.700183814

Reorder the terms:
0.375 + x = -1.700183814

Solving
0.375 + x = -1.700183814

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = -1.700183814 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -1.700183814 + -0.375
x = -1.700183814 + -0.375

Combine like terms: -1.700183814 + -0.375 = -2.075183814
x = -2.075183814

Simplifying
x = -2.075183814

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.325183814, -2.075183814}
Solution
x = {1.325183814, -2.075183814}