y^2+10y=19

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Solution for y^2+10y=19 equation:


Simplifying
y2 + 10y = 19

Reorder the terms:
10y + y2 = 19

Solving
10y + y2 = 19

Solving for variable 'y'.

Reorder the terms:
-19 + 10y + y2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 10y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 10y + 19 + y2 = 0 + 19

Reorder the terms:
-19 + 19 + 10y + y2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 10y + y2 = 0 + 19
10y + y2 = 0 + 19

Combine like terms: 0 + 19 = 19
10y + y2 = 19

The y term is 10y.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10y + 25 + y2 = 19 + 25

Reorder the terms:
25 + 10y + y2 = 19 + 25

Combine like terms: 19 + 25 = 44
25 + 10y + y2 = 44

Factor a perfect square on the left side:
(y + 5)(y + 5) = 44

Calculate the square root of the right side: 6.633249581

Break this problem into two subproblems by setting 
(y + 5) equal to 6.633249581 and -6.633249581.

Subproblem 1

y + 5 = 6.633249581 Simplifying y + 5 = 6.633249581 Reorder the terms: 5 + y = 6.633249581 Solving 5 + y = 6.633249581 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + y = 6.633249581 + -5 Combine like terms: 5 + -5 = 0 0 + y = 6.633249581 + -5 y = 6.633249581 + -5 Combine like terms: 6.633249581 + -5 = 1.633249581 y = 1.633249581 Simplifying y = 1.633249581

Subproblem 2

y + 5 = -6.633249581 Simplifying y + 5 = -6.633249581 Reorder the terms: 5 + y = -6.633249581 Solving 5 + y = -6.633249581 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + y = -6.633249581 + -5 Combine like terms: 5 + -5 = 0 0 + y = -6.633249581 + -5 y = -6.633249581 + -5 Combine like terms: -6.633249581 + -5 = -11.633249581 y = -11.633249581 Simplifying y = -11.633249581

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.633249581, -11.633249581}

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