4y^2-y-2=0

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


Simplifying
4y2 + -1y + -2 = 0

Reorder the terms:
-2 + -1y + 4y2 = 0

Solving
-2 + -1y + 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'.
-0.5 + -0.25y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.25y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.25y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.25y + y2 = 0 + 0.5
-0.25y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.25y + y2 = 0.5

The y term is -0.25y.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25y + 0.015625 + y2 = 0.5 + 0.015625

Reorder the terms:
0.015625 + -0.25y + y2 = 0.5 + 0.015625

Combine like terms: 0.5 + 0.015625 = 0.515625
0.015625 + -0.25y + y2 = 0.515625

Factor a perfect square on the left side:
(y + -0.125)(y + -0.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(y + -0.125) equal to 0.718070331 and -0.718070331.

Subproblem 1

y + -0.125 = 0.718070331 Simplifying y + -0.125 = 0.718070331 Reorder the terms: -0.125 + y = 0.718070331 Solving -0.125 + y = 0.718070331 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + y = 0.718070331 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + y = 0.718070331 + 0.125 y = 0.718070331 + 0.125 Combine like terms: 0.718070331 + 0.125 = 0.843070331 y = 0.843070331 Simplifying y = 0.843070331

Subproblem 2

y + -0.125 = -0.718070331 Simplifying y + -0.125 = -0.718070331 Reorder the terms: -0.125 + y = -0.718070331 Solving -0.125 + y = -0.718070331 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + y = -0.718070331 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + y = -0.718070331 + 0.125 y = -0.718070331 + 0.125 Combine like terms: -0.718070331 + 0.125 = -0.593070331 y = -0.593070331 Simplifying y = -0.593070331

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.843070331, -0.593070331}

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