4y^2-6y-19=0

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Solution for 4y^2-6y-19=0 equation:


Simplifying
4y2 + -6y + -19 = 0

Reorder the terms:
-19 + -6y + 4y2 = 0

Solving
-19 + -6y + 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'.
-4.75 + -1.5y + y2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + -1.5y + 4.75 + y2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + -1.5y + y2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + -1.5y + y2 = 0 + 4.75
-1.5y + y2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
-1.5y + y2 = 4.75

The y term is -1.5y.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5y + 0.5625 + y2 = 4.75 + 0.5625

Reorder the terms:
0.5625 + -1.5y + y2 = 4.75 + 0.5625

Combine like terms: 4.75 + 0.5625 = 5.3125
0.5625 + -1.5y + y2 = 5.3125

Factor a perfect square on the left side:
(y + -0.75)(y + -0.75) = 5.3125

Calculate the square root of the right side: 2.304886114

Break this problem into two subproblems by setting 
(y + -0.75) equal to 2.304886114 and -2.304886114.

Subproblem 1

y + -0.75 = 2.304886114 Simplifying y + -0.75 = 2.304886114 Reorder the terms: -0.75 + y = 2.304886114 Solving -0.75 + y = 2.304886114 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 2.304886114 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 2.304886114 + 0.75 y = 2.304886114 + 0.75 Combine like terms: 2.304886114 + 0.75 = 3.054886114 y = 3.054886114 Simplifying y = 3.054886114

Subproblem 2

y + -0.75 = -2.304886114 Simplifying y + -0.75 = -2.304886114 Reorder the terms: -0.75 + y = -2.304886114 Solving -0.75 + y = -2.304886114 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -2.304886114 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -2.304886114 + 0.75 y = -2.304886114 + 0.75 Combine like terms: -2.304886114 + 0.75 = -1.554886114 y = -1.554886114 Simplifying y = -1.554886114

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.054886114, -1.554886114}

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