9y^2+42y-5=0

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Solution for 9y^2+42y-5=0 equation: 


Simplifying
9y2 + 42y + -5 = 0

Reorder the terms:
-5 + 42y + 9y2 = 0

Solving
-5 + 42y + 9y2 = 0

Solving for variable 'y'.

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

Divide each side by '9'.
-0.5555555556 + 4.666666667y + y2 = 0

Move the constant term to the right:

Add '0.5555555556' to each side of the equation.
-0.5555555556 + 4.666666667y + 0.5555555556 + y2 = 0 + 0.5555555556

Reorder the terms:
-0.5555555556 + 0.5555555556 + 4.666666667y + y2 = 0 + 0.5555555556

Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000
0.0000000000 + 4.666666667y + y2 = 0 + 0.5555555556
4.666666667y + y2 = 0 + 0.5555555556

Combine like terms: 0 + 0.5555555556 = 0.5555555556
4.666666667y + y2 = 0.5555555556

The y term is 4.666666667y.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667y + 5.444444448 + y2 = 0.5555555556 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667y + y2 = 0.5555555556 + 5.444444448

Combine like terms: 0.5555555556 + 5.444444448 = 6.0000000036
5.444444448 + 4.666666667y + y2 = 6.0000000036

Factor a perfect square on the left side:
(y + 2.333333334)(y + 2.333333334) = 6.0000000036

Calculate the square root of the right side: 2.449489744

Break this problem into two subproblems by setting 
(y + 2.333333334) equal to 2.449489744 and -2.449489744.


Subproblem 1
y + 2.333333334 = 2.449489744

Simplifying
y + 2.333333334 = 2.449489744

Reorder the terms:
2.333333334 + y = 2.449489744

Solving
2.333333334 + y = 2.449489744

Solving for variable 'y'.

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

Add '-2.333333334' to each side of the equation.
2.333333334 + -2.333333334 + y = 2.449489744 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + y = 2.449489744 + -2.333333334
y = 2.449489744 + -2.333333334

Combine like terms: 2.449489744 + -2.333333334 = 0.11615641
y = 0.11615641

Simplifying
y = 0.11615641


Subproblem 2
y + 2.333333334 = -2.449489744

Simplifying
y + 2.333333334 = -2.449489744

Reorder the terms:
2.333333334 + y = -2.449489744

Solving
2.333333334 + y = -2.449489744

Solving for variable 'y'.

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

Add '-2.333333334' to each side of the equation.
2.333333334 + -2.333333334 + y = -2.449489744 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + y = -2.449489744 + -2.333333334
y = -2.449489744 + -2.333333334

Combine like terms: -2.449489744 + -2.333333334 = -4.782823078
y = -4.782823078

Simplifying
y = -4.782823078


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.11615641, -4.782823078}

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