4y^2-4y-23=0

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Solution for 4y^2-4y-23=0 equation:


Simplifying
4y2 + -4y + -23 = 0

Reorder the terms:
-23 + -4y + 4y2 = 0

Solving
-23 + -4y + 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'.
-5.75 + -1y + y2 = 0

Move the constant term to the right:

Add '5.75' to each side of the equation.
-5.75 + -1y + 5.75 + y2 = 0 + 5.75

Reorder the terms:
-5.75 + 5.75 + -1y + y2 = 0 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + -1y + y2 = 0 + 5.75
-1y + y2 = 0 + 5.75

Combine like terms: 0 + 5.75 = 5.75
-1y + y2 = 5.75

The y term is -1y.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1y + 0.25 + y2 = 5.75 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 5.75 + 0.25

Combine like terms: 5.75 + 0.25 = 6
0.25 + -1y + y2 = 6

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(y + -0.5) equal to 2.449489743 and -2.449489743.

Subproblem 1

y + -0.5 = 2.449489743 Simplifying y + -0.5 = 2.449489743 Reorder the terms: -0.5 + y = 2.449489743 Solving -0.5 + y = 2.449489743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = 2.449489743 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 2.449489743 + 0.5 y = 2.449489743 + 0.5 Combine like terms: 2.449489743 + 0.5 = 2.949489743 y = 2.949489743 Simplifying y = 2.949489743

Subproblem 2

y + -0.5 = -2.449489743 Simplifying y + -0.5 = -2.449489743 Reorder the terms: -0.5 + y = -2.449489743 Solving -0.5 + y = -2.449489743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = -2.449489743 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -2.449489743 + 0.5 y = -2.449489743 + 0.5 Combine like terms: -2.449489743 + 0.5 = -1.949489743 y = -1.949489743 Simplifying y = -1.949489743

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.949489743, -1.949489743}

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