4y^2-5y-30=0

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Solution for 4y^2-5y-30=0 equation:


Simplifying
4y2 + -5y + -30 = 0

Reorder the terms:
-30 + -5y + 4y2 = 0

Solving
-30 + -5y + 4y2 = 0

Solving for variable 'y'.

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

Divide each side by '4'.
-7.5 + -1.25y + y2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -1.25y + 7.5 + y2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -1.25y + y2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -1.25y + y2 = 0 + 7.5
-1.25y + y2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-1.25y + y2 = 7.5

The y term is -1.25y.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25y + 0.390625 + y2 = 7.5 + 0.390625

Reorder the terms:
0.390625 + -1.25y + y2 = 7.5 + 0.390625

Combine like terms: 7.5 + 0.390625 = 7.890625
0.390625 + -1.25y + y2 = 7.890625

Factor a perfect square on the left side:
(y + -0.625)(y + -0.625) = 7.890625

Calculate the square root of the right side: 2.809025632

Break this problem into two subproblems by setting 
(y + -0.625) equal to 2.809025632 and -2.809025632.

Subproblem 1

y + -0.625 = 2.809025632 Simplifying y + -0.625 = 2.809025632 Reorder the terms: -0.625 + y = 2.809025632 Solving -0.625 + y = 2.809025632 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + y = 2.809025632 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + y = 2.809025632 + 0.625 y = 2.809025632 + 0.625 Combine like terms: 2.809025632 + 0.625 = 3.434025632 y = 3.434025632 Simplifying y = 3.434025632

Subproblem 2

y + -0.625 = -2.809025632 Simplifying y + -0.625 = -2.809025632 Reorder the terms: -0.625 + y = -2.809025632 Solving -0.625 + y = -2.809025632 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + y = -2.809025632 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + y = -2.809025632 + 0.625 y = -2.809025632 + 0.625 Combine like terms: -2.809025632 + 0.625 = -2.184025632 y = -2.184025632 Simplifying y = -2.184025632

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.434025632, -2.184025632}

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