4.5k^2+4k-1.5=0

Solution for 4.5k^2+4k-1.5=0 equation:

Simplifying
4.5k2 + 4k + -1.5 = 0

Reorder the terms:
-1.5 + 4k + 4.5k2 = 0

Solving
-1.5 + 4k + 4.5k2 = 0

Solving for variable 'k'.

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

Divide each side by '4.5'.
-0.3333333333 + 0.8888888889k + k2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.8888888889k + 0.3333333333 + k2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.8888888889k + k2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.8888888889k + k2 = 0 + 0.3333333333
0.8888888889k + k2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.8888888889k + k2 = 0.3333333333

The k term is 0.8888888889k.  Take half its coefficient (0.4444444445).
Square it (0.1975308642) and add it to both sides.

Add '0.1975308642' to each side of the equation.
0.8888888889k + 0.1975308642 + k2 = 0.3333333333 + 0.1975308642

Reorder the terms:
0.1975308642 + 0.8888888889k + k2 = 0.3333333333 + 0.1975308642

Combine like terms: 0.3333333333 + 0.1975308642 = 0.5308641975
0.1975308642 + 0.8888888889k + k2 = 0.5308641975

Factor a perfect square on the left side:
(k + 0.4444444445)(k + 0.4444444445) = 0.5308641975

Calculate the square root of the right side: 0.72860428

Break this problem into two subproblems by setting 
(k + 0.4444444445) equal to 0.72860428 and -0.72860428.

Subproblem 1
k + 0.4444444445 = 0.72860428

Simplifying
k + 0.4444444445 = 0.72860428

Reorder the terms:
0.4444444445 + k = 0.72860428

Solving
0.4444444445 + k = 0.72860428

Solving for variable 'k'.

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

Add '-0.4444444445' to each side of the equation.
0.4444444445 + -0.4444444445 + k = 0.72860428 + -0.4444444445

Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000
0.0000000000 + k = 0.72860428 + -0.4444444445
k = 0.72860428 + -0.4444444445

Combine like terms: 0.72860428 + -0.4444444445 = 0.2841598355
k = 0.2841598355

Simplifying
k = 0.2841598355

Subproblem 2
k + 0.4444444445 = -0.72860428

Simplifying
k + 0.4444444445 = -0.72860428

Reorder the terms:
0.4444444445 + k = -0.72860428

Solving
0.4444444445 + k = -0.72860428

Solving for variable 'k'.

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

Add '-0.4444444445' to each side of the equation.
0.4444444445 + -0.4444444445 + k = -0.72860428 + -0.4444444445

Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000
0.0000000000 + k = -0.72860428 + -0.4444444445
k = -0.72860428 + -0.4444444445

Combine like terms: -0.72860428 + -0.4444444445 = -1.1730487245
k = -1.1730487245

Simplifying
k = -1.1730487245

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {0.2841598355, -1.1730487245}