y^2+19y=78

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Solution for y^2+19y=78 equation:


Simplifying
y2 + 19y = 78

Reorder the terms:
19y + y2 = 78

Solving
19y + y2 = 78

Solving for variable 'y'.

Reorder the terms:
-78 + 19y + y2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 19y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 19y + 78 + y2 = 0 + 78

Reorder the terms:
-78 + 78 + 19y + y2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 19y + y2 = 0 + 78
19y + y2 = 0 + 78

Combine like terms: 0 + 78 = 78
19y + y2 = 78

The y term is 19y.  Take half its coefficient (9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
19y + 90.25 + y2 = 78 + 90.25

Reorder the terms:
90.25 + 19y + y2 = 78 + 90.25

Combine like terms: 78 + 90.25 = 168.25
90.25 + 19y + y2 = 168.25

Factor a perfect square on the left side:
(y + 9.5)(y + 9.5) = 168.25

Calculate the square root of the right side: 12.971121771

Break this problem into two subproblems by setting 
(y + 9.5) equal to 12.971121771 and -12.971121771.

Subproblem 1

y + 9.5 = 12.971121771 Simplifying y + 9.5 = 12.971121771 Reorder the terms: 9.5 + y = 12.971121771 Solving 9.5 + y = 12.971121771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + y = 12.971121771 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + y = 12.971121771 + -9.5 y = 12.971121771 + -9.5 Combine like terms: 12.971121771 + -9.5 = 3.471121771 y = 3.471121771 Simplifying y = 3.471121771

Subproblem 2

y + 9.5 = -12.971121771 Simplifying y + 9.5 = -12.971121771 Reorder the terms: 9.5 + y = -12.971121771 Solving 9.5 + y = -12.971121771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + y = -12.971121771 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + y = -12.971121771 + -9.5 y = -12.971121771 + -9.5 Combine like terms: -12.971121771 + -9.5 = -22.471121771 y = -22.471121771 Simplifying y = -22.471121771

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.471121771, -22.471121771}

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