45x(4x+3)=1000

Solution for 45x(4x+3)=1000 equation:

Simplifying
45x(4x + 3) = 1000

Reorder the terms:
45x(3 + 4x) = 1000
(3 * 45x + 4x * 45x) = 1000
(135x + 180x2) = 1000

Solving
135x + 180x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + 135x + 180x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 135x + 180x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-200 + 27x + 36x2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-200 + 27x + 36x2)' equal to zero and attempt to solve:

Simplifying
-200 + 27x + 36x2 = 0

Solving
-200 + 27x + 36x2 = 0

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

Divide each side by '36'.
-5.555555556 + 0.75x + x2 = 0

Move the constant term to the right:

Add '5.555555556' to each side of the equation.
-5.555555556 + 0.75x + 5.555555556 + x2 = 0 + 5.555555556

Reorder the terms:
-5.555555556 + 5.555555556 + 0.75x + x2 = 0 + 5.555555556

Combine like terms: -5.555555556 + 5.555555556 = 0.000000000
0.000000000 + 0.75x + x2 = 0 + 5.555555556
0.75x + x2 = 0 + 5.555555556

Combine like terms: 0 + 5.555555556 = 5.555555556
0.75x + x2 = 5.555555556

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 = 5.555555556 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 5.555555556 + 0.140625

Combine like terms: 5.555555556 + 0.140625 = 5.696180556
0.140625 + 0.75x + x2 = 5.696180556

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 5.696180556

Calculate the square root of the right side: 2.386667249

Break this problem into two subproblems by setting 
(x + 0.375) equal to 2.386667249 and -2.386667249.

Subproblem 1
x + 0.375 = 2.386667249

Simplifying
x + 0.375 = 2.386667249

Reorder the terms:
0.375 + x = 2.386667249

Solving
0.375 + x = 2.386667249

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 = 2.386667249 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 2.386667249 + -0.375
x = 2.386667249 + -0.375

Combine like terms: 2.386667249 + -0.375 = 2.011667249
x = 2.011667249

Simplifying
x = 2.011667249

Subproblem 2
x + 0.375 = -2.386667249

Simplifying
x + 0.375 = -2.386667249

Reorder the terms:
0.375 + x = -2.386667249

Solving
0.375 + x = -2.386667249

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 = -2.386667249 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -2.386667249 + -0.375
x = -2.386667249 + -0.375

Combine like terms: -2.386667249 + -0.375 = -2.761667249
x = -2.761667249

Simplifying
x = -2.761667249

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.011667249, -2.761667249}
Solution
x = {2.011667249, -2.761667249}