(2y^2)+7y-78=0

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Solution for (2y^2)+7y-78=0 equation:


Simplifying
(2y2) + 7y + -78 = 0

Reorder the terms:
-78 + 7y + (2y2) = 0

Solving
-78 + 7y + (2y2) = 0

Solving for variable 'y'.

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

Divide each side by '2'.
-39 + 3.5y + y2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 3.5y + 39 + y2 = 0 + 39

Reorder the terms:
-39 + 39 + 3.5y + y2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 3.5y + y2 = 0 + 39
3.5y + y2 = 0 + 39

Combine like terms: 0 + 39 = 39
3.5y + y2 = 39

The y term is 3.5y.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5y + 3.0625 + y2 = 39 + 3.0625

Reorder the terms:
3.0625 + 3.5y + y2 = 39 + 3.0625

Combine like terms: 39 + 3.0625 = 42.0625
3.0625 + 3.5y + y2 = 42.0625

Factor a perfect square on the left side:
((y) + 1.75)((y) + 1.75) = 42.0625

Calculate the square root of the right side: 6.485560886

Break this problem into two subproblems by setting 
((y) + 1.75) equal to 6.485560886 and -6.485560886.

Subproblem 1

(y) + 1.75 = 6.485560886 Simplifying (y) + 1.75 = 6.485560886 y + 1.75 = 6.485560886 Reorder the terms: 1.75 + y = 6.485560886 Solving 1.75 + y = 6.485560886 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = 6.485560886 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = 6.485560886 + -1.75 y = 6.485560886 + -1.75 Combine like terms: 6.485560886 + -1.75 = 4.735560886 y = 4.735560886 Simplifying y = 4.735560886

Subproblem 2

(y) + 1.75 = -6.485560886 Simplifying (y) + 1.75 = -6.485560886 y + 1.75 = -6.485560886 Reorder the terms: 1.75 + y = -6.485560886 Solving 1.75 + y = -6.485560886 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = -6.485560886 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = -6.485560886 + -1.75 y = -6.485560886 + -1.75 Combine like terms: -6.485560886 + -1.75 = -8.235560886 y = -8.235560886 Simplifying y = -8.235560886

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.735560886, -8.235560886}

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