0=3y^2-19y+22

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Solution for 0=3y^2-19y+22 equation:


Simplifying
0 = 3y2 + -19y + 22

Reorder the terms:
0 = 22 + -19y + 3y2

Solving
0 = 22 + -19y + 3y2

Solving for variable 'y'.

Combine like terms: 0 + -22 = -22
-22 + 19y + -3y2 = 22 + -19y + 3y2 + -22 + 19y + -3y2

Reorder the terms:
-22 + 19y + -3y2 = 22 + -22 + -19y + 19y + 3y2 + -3y2

Combine like terms: 22 + -22 = 0
-22 + 19y + -3y2 = 0 + -19y + 19y + 3y2 + -3y2
-22 + 19y + -3y2 = -19y + 19y + 3y2 + -3y2

Combine like terms: -19y + 19y = 0
-22 + 19y + -3y2 = 0 + 3y2 + -3y2
-22 + 19y + -3y2 = 3y2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
-22 + 19y + -3y2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
7.333333333 + -6.333333333y + y2 = 0

Move the constant term to the right:

Add '-7.333333333' to each side of the equation.
7.333333333 + -6.333333333y + -7.333333333 + y2 = 0 + -7.333333333

Reorder the terms:
7.333333333 + -7.333333333 + -6.333333333y + y2 = 0 + -7.333333333

Combine like terms: 7.333333333 + -7.333333333 = 0.000000000
0.000000000 + -6.333333333y + y2 = 0 + -7.333333333
-6.333333333y + y2 = 0 + -7.333333333

Combine like terms: 0 + -7.333333333 = -7.333333333
-6.333333333y + y2 = -7.333333333

The y term is -6.333333333y.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333y + 10.02777778 + y2 = -7.333333333 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333y + y2 = -7.333333333 + 10.02777778

Combine like terms: -7.333333333 + 10.02777778 = 2.694444447
10.02777778 + -6.333333333y + y2 = 2.694444447

Factor a perfect square on the left side:
(y + -3.166666667)(y + -3.166666667) = 2.694444447

Calculate the square root of the right side: 1.641476301

Break this problem into two subproblems by setting 
(y + -3.166666667) equal to 1.641476301 and -1.641476301.

Subproblem 1

y + -3.166666667 = 1.641476301 Simplifying y + -3.166666667 = 1.641476301 Reorder the terms: -3.166666667 + y = 1.641476301 Solving -3.166666667 + y = 1.641476301 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + y = 1.641476301 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + y = 1.641476301 + 3.166666667 y = 1.641476301 + 3.166666667 Combine like terms: 1.641476301 + 3.166666667 = 4.808142968 y = 4.808142968 Simplifying y = 4.808142968

Subproblem 2

y + -3.166666667 = -1.641476301 Simplifying y + -3.166666667 = -1.641476301 Reorder the terms: -3.166666667 + y = -1.641476301 Solving -3.166666667 + y = -1.641476301 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + y = -1.641476301 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + y = -1.641476301 + 3.166666667 y = -1.641476301 + 3.166666667 Combine like terms: -1.641476301 + 3.166666667 = 1.525190366 y = 1.525190366 Simplifying y = 1.525190366

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.808142968, 1.525190366}

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