45y^2-130y+84=0

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Solution for 45y^2-130y+84=0 equation:


Simplifying
45y2 + -130y + 84 = 0

Reorder the terms:
84 + -130y + 45y2 = 0

Solving
84 + -130y + 45y2 = 0

Solving for variable 'y'.

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

Divide each side by '45'.
1.866666667 + -2.888888889y + y2 = 0

Move the constant term to the right:

Add '-1.866666667' to each side of the equation.
1.866666667 + -2.888888889y + -1.866666667 + y2 = 0 + -1.866666667

Reorder the terms:
1.866666667 + -1.866666667 + -2.888888889y + y2 = 0 + -1.866666667

Combine like terms: 1.866666667 + -1.866666667 = 0.000000000
0.000000000 + -2.888888889y + y2 = 0 + -1.866666667
-2.888888889y + y2 = 0 + -1.866666667

Combine like terms: 0 + -1.866666667 = -1.866666667
-2.888888889y + y2 = -1.866666667

The y term is -2.888888889y.  Take half its coefficient (-1.444444445).
Square it (2.086419755) and add it to both sides.

Add '2.086419755' to each side of the equation.
-2.888888889y + 2.086419755 + y2 = -1.866666667 + 2.086419755

Reorder the terms:
2.086419755 + -2.888888889y + y2 = -1.866666667 + 2.086419755

Combine like terms: -1.866666667 + 2.086419755 = 0.219753088
2.086419755 + -2.888888889y + y2 = 0.219753088

Factor a perfect square on the left side:
(y + -1.444444445)(y + -1.444444445) = 0.219753088

Calculate the square root of the right side: 0.468778293

Break this problem into two subproblems by setting 
(y + -1.444444445) equal to 0.468778293 and -0.468778293.

Subproblem 1

y + -1.444444445 = 0.468778293 Simplifying y + -1.444444445 = 0.468778293 Reorder the terms: -1.444444445 + y = 0.468778293 Solving -1.444444445 + y = 0.468778293 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + y = 0.468778293 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + y = 0.468778293 + 1.444444445 y = 0.468778293 + 1.444444445 Combine like terms: 0.468778293 + 1.444444445 = 1.913222738 y = 1.913222738 Simplifying y = 1.913222738

Subproblem 2

y + -1.444444445 = -0.468778293 Simplifying y + -1.444444445 = -0.468778293 Reorder the terms: -1.444444445 + y = -0.468778293 Solving -1.444444445 + y = -0.468778293 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + y = -0.468778293 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + y = -0.468778293 + 1.444444445 y = -0.468778293 + 1.444444445 Combine like terms: -0.468778293 + 1.444444445 = 0.975666152 y = 0.975666152 Simplifying y = 0.975666152

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.913222738, 0.975666152}

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