2y^2-24+3y=12-18

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Solution for 2y^2-24+3y=12-18 equation:


Simplifying
2y2 + -24 + 3y = 12 + -18

Reorder the terms:
-24 + 3y + 2y2 = 12 + -18

Combine like terms: 12 + -18 = -6
-24 + 3y + 2y2 = -6

Solving
-24 + 3y + 2y2 = -6

Solving for variable 'y'.

Reorder the terms:
-24 + 6 + 3y + 2y2 = -6 + 6

Combine like terms: -24 + 6 = -18
-18 + 3y + 2y2 = -6 + 6

Combine like terms: -6 + 6 = 0
-18 + 3y + 2y2 = 0

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

Divide each side by '2'.
-9 + 1.5y + y2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 1.5y + 9 + y2 = 0 + 9

Reorder the terms:
-9 + 9 + 1.5y + y2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 1.5y + y2 = 0 + 9
1.5y + y2 = 0 + 9

Combine like terms: 0 + 9 = 9
1.5y + y2 = 9

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 = 9 + 0.5625

Reorder the terms:
0.5625 + 1.5y + y2 = 9 + 0.5625

Combine like terms: 9 + 0.5625 = 9.5625
0.5625 + 1.5y + y2 = 9.5625

Factor a perfect square on the left side:
(y + 0.75)(y + 0.75) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(y + 0.75) equal to 3.092329219 and -3.092329219.

Subproblem 1

y + 0.75 = 3.092329219 Simplifying y + 0.75 = 3.092329219 Reorder the terms: 0.75 + y = 3.092329219 Solving 0.75 + y = 3.092329219 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 = 3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 3.092329219 + -0.75 y = 3.092329219 + -0.75 Combine like terms: 3.092329219 + -0.75 = 2.342329219 y = 2.342329219 Simplifying y = 2.342329219

Subproblem 2

y + 0.75 = -3.092329219 Simplifying y + 0.75 = -3.092329219 Reorder the terms: 0.75 + y = -3.092329219 Solving 0.75 + y = -3.092329219 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 = -3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -3.092329219 + -0.75 y = -3.092329219 + -0.75 Combine like terms: -3.092329219 + -0.75 = -3.842329219 y = -3.842329219 Simplifying y = -3.842329219

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.342329219, -3.842329219}

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