4y^2-20y+23=0

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Solution for 4y^2-20y+23=0 equation:


Simplifying
4y2 + -20y + 23 = 0

Reorder the terms:
23 + -20y + 4y2 = 0

Solving
23 + -20y + 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 + -5y + y2 = 0

Move the constant term to the right:

Add '-5.75' to each side of the equation.
5.75 + -5y + -5.75 + y2 = 0 + -5.75

Reorder the terms:
5.75 + -5.75 + -5y + y2 = 0 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + -5y + y2 = 0 + -5.75
-5y + y2 = 0 + -5.75

Combine like terms: 0 + -5.75 = -5.75
-5y + y2 = -5.75

The y term is -5y.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5y + 6.25 + y2 = -5.75 + 6.25

Reorder the terms:
6.25 + -5y + y2 = -5.75 + 6.25

Combine like terms: -5.75 + 6.25 = 0.5
6.25 + -5y + y2 = 0.5

Factor a perfect square on the left side:
(y + -2.5)(y + -2.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(y + -2.5) equal to 0.707106781 and -0.707106781.

Subproblem 1

y + -2.5 = 0.707106781 Simplifying y + -2.5 = 0.707106781 Reorder the terms: -2.5 + y = 0.707106781 Solving -2.5 + y = 0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = 0.707106781 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = 0.707106781 + 2.5 y = 0.707106781 + 2.5 Combine like terms: 0.707106781 + 2.5 = 3.207106781 y = 3.207106781 Simplifying y = 3.207106781

Subproblem 2

y + -2.5 = -0.707106781 Simplifying y + -2.5 = -0.707106781 Reorder the terms: -2.5 + y = -0.707106781 Solving -2.5 + y = -0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = -0.707106781 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = -0.707106781 + 2.5 y = -0.707106781 + 2.5 Combine like terms: -0.707106781 + 2.5 = 1.792893219 y = 1.792893219 Simplifying y = 1.792893219

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.207106781, 1.792893219}

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