14y+22-15y^2+23y=

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Solution for 14y+22-15y^2+23y= equation: 


Simplifying
14y + 22 + -15y2 + 23y = 0

Reorder the terms:
22 + 14y + 23y + -15y2 = 0

Combine like terms: 14y + 23y = 37y
22 + 37y + -15y2 = 0

Solving
22 + 37y + -15y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
-15 the coefficient of the squared term: 

Divide each side by '-15'.
-1.466666667 + -2.466666667y + y2 = 0

Move the constant term to the right:

Add '1.466666667' to each side of the equation.
-1.466666667 + -2.466666667y + 1.466666667 + y2 = 0 + 1.466666667

Reorder the terms:
-1.466666667 + 1.466666667 + -2.466666667y + y2 = 0 + 1.466666667

Combine like terms: -1.466666667 + 1.466666667 = 0.000000000
0.000000000 + -2.466666667y + y2 = 0 + 1.466666667
-2.466666667y + y2 = 0 + 1.466666667

Combine like terms: 0 + 1.466666667 = 1.466666667
-2.466666667y + y2 = 1.466666667

The y term is -2.466666667y.  Take half its coefficient (-1.233333334).
Square it (1.521111113) and add it to both sides.

Add '1.521111113' to each side of the equation.
-2.466666667y + 1.521111113 + y2 = 1.466666667 + 1.521111113

Reorder the terms:
1.521111113 + -2.466666667y + y2 = 1.466666667 + 1.521111113

Combine like terms: 1.466666667 + 1.521111113 = 2.98777778
1.521111113 + -2.466666667y + y2 = 2.98777778

Factor a perfect square on the left side:
(y + -1.233333334)(y + -1.233333334) = 2.98777778

Calculate the square root of the right side: 1.728518956

Break this problem into two subproblems by setting 
(y + -1.233333334) equal to 1.728518956 and -1.728518956.


Subproblem 1
y + -1.233333334 = 1.728518956

Simplifying
y + -1.233333334 = 1.728518956

Reorder the terms:
-1.233333334 + y = 1.728518956

Solving
-1.233333334 + y = 1.728518956

Solving for variable 'y'.

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

Add '1.233333334' to each side of the equation.
-1.233333334 + 1.233333334 + y = 1.728518956 + 1.233333334

Combine like terms: -1.233333334 + 1.233333334 = 0.000000000
0.000000000 + y = 1.728518956 + 1.233333334
y = 1.728518956 + 1.233333334

Combine like terms: 1.728518956 + 1.233333334 = 2.96185229
y = 2.96185229

Simplifying
y = 2.96185229


Subproblem 2
y + -1.233333334 = -1.728518956

Simplifying
y + -1.233333334 = -1.728518956

Reorder the terms:
-1.233333334 + y = -1.728518956

Solving
-1.233333334 + y = -1.728518956

Solving for variable 'y'.

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

Add '1.233333334' to each side of the equation.
-1.233333334 + 1.233333334 + y = -1.728518956 + 1.233333334

Combine like terms: -1.233333334 + 1.233333334 = 0.000000000
0.000000000 + y = -1.728518956 + 1.233333334
y = -1.728518956 + 1.233333334

Combine like terms: -1.728518956 + 1.233333334 = -0.495185622
y = -0.495185622

Simplifying
y = -0.495185622


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.96185229, -0.495185622}

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