y+1.5y-0.2=4.2y(y-4.5)

Solution for y+1.5y-0.2=4.2y(y-4.5) equation:

Simplifying
y + 1.5y + -0.2 = 4.2y(y + -4.5)

Reorder the terms:
-0.2 + y + 1.5y = 4.2y(y + -4.5)

Combine like terms: y + 1.5y = 2.5y
-0.2 + 2.5y = 4.2y(y + -4.5)

Reorder the terms:
-0.2 + 2.5y = 4.2y(-4.5 + y)
-0.2 + 2.5y = (-4.5 * 4.2y + y * 4.2y)
-0.2 + 2.5y = (-18.9y + 4.2y2)

Solving
-0.2 + 2.5y = -18.9y + 4.2y2

Solving for variable 'y'.

Combine like terms: 2.5y + 18.9y = 21.4y
-0.2 + 21.4y + -4.2y2 = -18.9y + 4.2y2 + 18.9y + -4.2y2

Reorder the terms:
-0.2 + 21.4y + -4.2y2 = -18.9y + 18.9y + 4.2y2 + -4.2y2

Combine like terms: -18.9y + 18.9y = 0.0
-0.2 + 21.4y + -4.2y2 = 0.0 + 4.2y2 + -4.2y2
-0.2 + 21.4y + -4.2y2 = 4.2y2 + -4.2y2

Combine like terms: 4.2y2 + -4.2y2 = 0.0
-0.2 + 21.4y + -4.2y2 = 0.0

Begin completing the square.  Divide all terms by
-4.2 the coefficient of the squared term: 

Divide each side by '-4.2'.
0.04761904762 + -5.095238095y + y2 = 0

Move the constant term to the right:

Add '-0.04761904762' to each side of the equation.
0.04761904762 + -5.095238095y + -0.04761904762 + y2 = 0 + -0.04761904762

Reorder the terms:
0.04761904762 + -0.04761904762 + -5.095238095y + y2 = 0 + -0.04761904762

Combine like terms: 0.04761904762 + -0.04761904762 = 0.00000000000
0.00000000000 + -5.095238095y + y2 = 0 + -0.04761904762
-5.095238095y + y2 = 0 + -0.04761904762

Combine like terms: 0 + -0.04761904762 = -0.04761904762
-5.095238095y + y2 = -0.04761904762

The y term is -5.095238095y.  Take half its coefficient (-2.547619048).
Square it (6.490362814) and add it to both sides.

Add '6.490362814' to each side of the equation.
-5.095238095y + 6.490362814 + y2 = -0.04761904762 + 6.490362814

Reorder the terms:
6.490362814 + -5.095238095y + y2 = -0.04761904762 + 6.490362814

Combine like terms: -0.04761904762 + 6.490362814 = 6.44274376638
6.490362814 + -5.095238095y + y2 = 6.44274376638

Factor a perfect square on the left side:
(y + -2.547619048)(y + -2.547619048) = 6.44274376638

Calculate the square root of the right side: 2.538256048

Break this problem into two subproblems by setting 
(y + -2.547619048) equal to 2.538256048 and -2.538256048.

Subproblem 1
y + -2.547619048 = 2.538256048

Simplifying
y + -2.547619048 = 2.538256048

Reorder the terms:
-2.547619048 + y = 2.538256048

Solving
-2.547619048 + y = 2.538256048

Solving for variable 'y'.

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

Add '2.547619048' to each side of the equation.
-2.547619048 + 2.547619048 + y = 2.538256048 + 2.547619048

Combine like terms: -2.547619048 + 2.547619048 = 0.000000000
0.000000000 + y = 2.538256048 + 2.547619048
y = 2.538256048 + 2.547619048

Combine like terms: 2.538256048 + 2.547619048 = 5.085875096
y = 5.085875096

Simplifying
y = 5.085875096

Subproblem 2
y + -2.547619048 = -2.538256048

Simplifying
y + -2.547619048 = -2.538256048

Reorder the terms:
-2.547619048 + y = -2.538256048

Solving
-2.547619048 + y = -2.538256048

Solving for variable 'y'.

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

Add '2.547619048' to each side of the equation.
-2.547619048 + 2.547619048 + y = -2.538256048 + 2.547619048

Combine like terms: -2.547619048 + 2.547619048 = 0.000000000
0.000000000 + y = -2.538256048 + 2.547619048
y = -2.538256048 + 2.547619048

Combine like terms: -2.538256048 + 2.547619048 = 0.009363
y = 0.009363

Simplifying
y = 0.009363

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.085875096, 0.009363}