4y^2+54y-9=0

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Solution for 4y^2+54y-9=0 equation:


Simplifying
4y2 + 54y + -9 = 0

Reorder the terms:
-9 + 54y + 4y2 = 0

Solving
-9 + 54y + 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'.
-2.25 + 13.5y + y2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + 13.5y + 2.25 + y2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + 13.5y + y2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + 13.5y + y2 = 0 + 2.25
13.5y + y2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
13.5y + y2 = 2.25

The y term is 13.5y.  Take half its coefficient (6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
13.5y + 45.5625 + y2 = 2.25 + 45.5625

Reorder the terms:
45.5625 + 13.5y + y2 = 2.25 + 45.5625

Combine like terms: 2.25 + 45.5625 = 47.8125
45.5625 + 13.5y + y2 = 47.8125

Factor a perfect square on the left side:
(y + 6.75)(y + 6.75) = 47.8125

Calculate the square root of the right side: 6.914658343

Break this problem into two subproblems by setting 
(y + 6.75) equal to 6.914658343 and -6.914658343.

Subproblem 1

y + 6.75 = 6.914658343 Simplifying y + 6.75 = 6.914658343 Reorder the terms: 6.75 + y = 6.914658343 Solving 6.75 + y = 6.914658343 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + y = 6.914658343 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + y = 6.914658343 + -6.75 y = 6.914658343 + -6.75 Combine like terms: 6.914658343 + -6.75 = 0.164658343 y = 0.164658343 Simplifying y = 0.164658343

Subproblem 2

y + 6.75 = -6.914658343 Simplifying y + 6.75 = -6.914658343 Reorder the terms: 6.75 + y = -6.914658343 Solving 6.75 + y = -6.914658343 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + y = -6.914658343 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + y = -6.914658343 + -6.75 y = -6.914658343 + -6.75 Combine like terms: -6.914658343 + -6.75 = -13.664658343 y = -13.664658343 Simplifying y = -13.664658343

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.164658343, -13.664658343}

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