6y+27=4y^2

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Solution for 6y+27=4y^2 equation:


Simplifying
6y + 27 = 4y2

Reorder the terms:
27 + 6y = 4y2

Solving
27 + 6y = 4y2

Solving for variable 'y'.

Combine like terms: 4y2 + -4y2 = 0
27 + 6y + -4y2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-6.75 + -1.5y + y2 = 0

Move the constant term to the right:

Add '6.75' to each side of the equation.
-6.75 + -1.5y + 6.75 + y2 = 0 + 6.75

Reorder the terms:
-6.75 + 6.75 + -1.5y + y2 = 0 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + -1.5y + y2 = 0 + 6.75
-1.5y + y2 = 0 + 6.75

Combine like terms: 0 + 6.75 = 6.75
-1.5y + y2 = 6.75

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 = 6.75 + 0.5625

Reorder the terms:
0.5625 + -1.5y + y2 = 6.75 + 0.5625

Combine like terms: 6.75 + 0.5625 = 7.3125
0.5625 + -1.5y + y2 = 7.3125

Factor a perfect square on the left side:
(y + -0.75)(y + -0.75) = 7.3125

Calculate the square root of the right side: 2.704163457

Break this problem into two subproblems by setting 
(y + -0.75) equal to 2.704163457 and -2.704163457.

Subproblem 1

y + -0.75 = 2.704163457 Simplifying y + -0.75 = 2.704163457 Reorder the terms: -0.75 + y = 2.704163457 Solving -0.75 + y = 2.704163457 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 = 2.704163457 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 2.704163457 + 0.75 y = 2.704163457 + 0.75 Combine like terms: 2.704163457 + 0.75 = 3.454163457 y = 3.454163457 Simplifying y = 3.454163457

Subproblem 2

y + -0.75 = -2.704163457 Simplifying y + -0.75 = -2.704163457 Reorder the terms: -0.75 + y = -2.704163457 Solving -0.75 + y = -2.704163457 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 = -2.704163457 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -2.704163457 + 0.75 y = -2.704163457 + 0.75 Combine like terms: -2.704163457 + 0.75 = -1.954163457 y = -1.954163457 Simplifying y = -1.954163457

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.454163457, -1.954163457}

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