X(4X+5)-3=0

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Solution for X(4X+5)-3=0 equation: 


Simplifying
X(4X + 5) + -3 = 0

Reorder the terms:
X(5 + 4X) + -3 = 0
(5 * X + 4X * X) + -3 = 0
(5X + 4X2) + -3 = 0

Reorder the terms:
-3 + 5X + 4X2 = 0

Solving
-3 + 5X + 4X2 = 0

Solving for variable 'X'.

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

Divide each side by '4'.
-0.75 + 1.25X + X2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 1.25X + 0.75 + X2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 1.25X + X2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 1.25X + X2 = 0 + 0.75
1.25X + X2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
1.25X + X2 = 0.75

The X term is 1.25X.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25X + 0.390625 + X2 = 0.75 + 0.390625

Reorder the terms:
0.390625 + 1.25X + X2 = 0.75 + 0.390625

Combine like terms: 0.75 + 0.390625 = 1.140625
0.390625 + 1.25X + X2 = 1.140625

Factor a perfect square on the left side:
(X + 0.625)(X + 0.625) = 1.140625

Calculate the square root of the right side: 1.068000468

Break this problem into two subproblems by setting 
(X + 0.625) equal to 1.068000468 and -1.068000468.


Subproblem 1
X + 0.625 = 1.068000468

Simplifying
X + 0.625 = 1.068000468

Reorder the terms:
0.625 + X = 1.068000468

Solving
0.625 + X = 1.068000468

Solving for variable 'X'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + X = 1.068000468 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + X = 1.068000468 + -0.625
X = 1.068000468 + -0.625

Combine like terms: 1.068000468 + -0.625 = 0.443000468
X = 0.443000468

Simplifying
X = 0.443000468


Subproblem 2
X + 0.625 = -1.068000468

Simplifying
X + 0.625 = -1.068000468

Reorder the terms:
0.625 + X = -1.068000468

Solving
0.625 + X = -1.068000468

Solving for variable 'X'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + X = -1.068000468 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + X = -1.068000468 + -0.625
X = -1.068000468 + -0.625

Combine like terms: -1.068000468 + -0.625 = -1.693000468
X = -1.693000468

Simplifying
X = -1.693000468


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {0.443000468, -1.693000468}

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