y^2+3y=78

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Solution for y^2+3y=78 equation:


Simplifying
y2 + 3y = 78

Reorder the terms:
3y + y2 = 78

Solving
3y + y2 = 78

Solving for variable 'y'.

Reorder the terms:
-78 + 3y + y2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 3y + 78 + y2 = 0 + 78

Reorder the terms:
-78 + 78 + 3y + y2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 3y + y2 = 0 + 78
3y + y2 = 0 + 78

Combine like terms: 0 + 78 = 78
3y + y2 = 78

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 78 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 78 + 2.25

Combine like terms: 78 + 2.25 = 80.25
2.25 + 3y + y2 = 80.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 80.25

Calculate the square root of the right side: 8.958236434

Break this problem into two subproblems by setting 
(y + 1.5) equal to 8.958236434 and -8.958236434.

Subproblem 1

y + 1.5 = 8.958236434 Simplifying y + 1.5 = 8.958236434 Reorder the terms: 1.5 + y = 8.958236434 Solving 1.5 + y = 8.958236434 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 8.958236434 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 8.958236434 + -1.5 y = 8.958236434 + -1.5 Combine like terms: 8.958236434 + -1.5 = 7.458236434 y = 7.458236434 Simplifying y = 7.458236434

Subproblem 2

y + 1.5 = -8.958236434 Simplifying y + 1.5 = -8.958236434 Reorder the terms: 1.5 + y = -8.958236434 Solving 1.5 + y = -8.958236434 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -8.958236434 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -8.958236434 + -1.5 y = -8.958236434 + -1.5 Combine like terms: -8.958236434 + -1.5 = -10.458236434 y = -10.458236434 Simplifying y = -10.458236434

Solution

The solution to the problem is based on the solutions from the subproblems. y = {7.458236434, -10.458236434}

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