y(2y-1)=49

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Solution for y(2y-1)=49 equation:


Simplifying
y(2y + -1) = 49

Reorder the terms:
y(-1 + 2y) = 49
(-1 * y + 2y * y) = 49
(-1y + 2y2) = 49

Solving
-1y + 2y2 = 49

Solving for variable 'y'.

Reorder the terms:
-49 + -1y + 2y2 = 49 + -49

Combine like terms: 49 + -49 = 0
-49 + -1y + 2y2 = 0

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

Divide each side by '2'.
-24.5 + -0.5y + y2 = 0

Move the constant term to the right:

Add '24.5' to each side of the equation.
-24.5 + -0.5y + 24.5 + y2 = 0 + 24.5

Reorder the terms:
-24.5 + 24.5 + -0.5y + y2 = 0 + 24.5

Combine like terms: -24.5 + 24.5 = 0.0
0.0 + -0.5y + y2 = 0 + 24.5
-0.5y + y2 = 0 + 24.5

Combine like terms: 0 + 24.5 = 24.5
-0.5y + y2 = 24.5

The y term is -0.5y.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5y + 0.0625 + y2 = 24.5 + 0.0625

Reorder the terms:
0.0625 + -0.5y + y2 = 24.5 + 0.0625

Combine like terms: 24.5 + 0.0625 = 24.5625
0.0625 + -0.5y + y2 = 24.5625

Factor a perfect square on the left side:
(y + -0.25)(y + -0.25) = 24.5625

Calculate the square root of the right side: 4.9560569

Break this problem into two subproblems by setting 
(y + -0.25) equal to 4.9560569 and -4.9560569.

Subproblem 1

y + -0.25 = 4.9560569 Simplifying y + -0.25 = 4.9560569 Reorder the terms: -0.25 + y = 4.9560569 Solving -0.25 + y = 4.9560569 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = 4.9560569 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = 4.9560569 + 0.25 y = 4.9560569 + 0.25 Combine like terms: 4.9560569 + 0.25 = 5.2060569 y = 5.2060569 Simplifying y = 5.2060569

Subproblem 2

y + -0.25 = -4.9560569 Simplifying y + -0.25 = -4.9560569 Reorder the terms: -0.25 + y = -4.9560569 Solving -0.25 + y = -4.9560569 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = -4.9560569 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = -4.9560569 + 0.25 y = -4.9560569 + 0.25 Combine like terms: -4.9560569 + 0.25 = -4.7060569 y = -4.7060569 Simplifying y = -4.7060569

Solution

The solution to the problem is based on the solutions from the subproblems. y = {5.2060569, -4.7060569}

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