y^2+1.5y=54

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Solution for y^2+1.5y=54 equation:


Simplifying
y2 + 1.5y = 54

Reorder the terms:
1.5y + y2 = 54

Solving
1.5y + y2 = 54

Solving for variable 'y'.

Reorder the terms:
-54 + 1.5y + y2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 1.5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 1.5y + 54 + y2 = 0 + 54

Reorder the terms:
-54 + 54 + 1.5y + y2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + 1.5y + y2 = 0 + 54
1.5y + y2 = 0 + 54

Combine like terms: 0 + 54 = 54
1.5y + y2 = 54

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 = 54 + 0.5625

Reorder the terms:
0.5625 + 1.5y + y2 = 54 + 0.5625

Combine like terms: 54 + 0.5625 = 54.5625
0.5625 + 1.5y + y2 = 54.5625

Factor a perfect square on the left side:
(y + 0.75)(y + 0.75) = 54.5625

Calculate the square root of the right side: 7.386643351

Break this problem into two subproblems by setting 
(y + 0.75) equal to 7.386643351 and -7.386643351.

Subproblem 1

y + 0.75 = 7.386643351 Simplifying y + 0.75 = 7.386643351 Reorder the terms: 0.75 + y = 7.386643351 Solving 0.75 + y = 7.386643351 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 = 7.386643351 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 7.386643351 + -0.75 y = 7.386643351 + -0.75 Combine like terms: 7.386643351 + -0.75 = 6.636643351 y = 6.636643351 Simplifying y = 6.636643351

Subproblem 2

y + 0.75 = -7.386643351 Simplifying y + 0.75 = -7.386643351 Reorder the terms: 0.75 + y = -7.386643351 Solving 0.75 + y = -7.386643351 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 = -7.386643351 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -7.386643351 + -0.75 y = -7.386643351 + -0.75 Combine like terms: -7.386643351 + -0.75 = -8.136643351 y = -8.136643351 Simplifying y = -8.136643351

Solution

The solution to the problem is based on the solutions from the subproblems. y = {6.636643351, -8.136643351}

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