3x(4x-5)=9

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Solution for 3x(4x-5)=9 equation: 


Simplifying
3x(4x + -5) = 9

Reorder the terms:
3x(-5 + 4x) = 9
(-5 * 3x + 4x * 3x) = 9
(-15x + 12x2) = 9

Solving
-15x + 12x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + -15x + 12x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + -15x + 12x2 = 0

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

Ignore the factor 3.

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

Simplifying
-3 + -5x + 4x2 = 0

Solving
-3 + -5x + 4x2 = 0

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 = 1.693000468
x = 1.693000468

Simplifying
x = 1.693000468


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 = -0.443000468
x = -0.443000468

Simplifying
x = -0.443000468


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

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