y^2+5y=7

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Solution for y^2+5y=7 equation: 


Simplifying
y2 + 5y = 7

Reorder the terms:
5y + y2 = 7

Solving
5y + y2 = 7

Solving for variable 'y'.

Reorder the terms:
-7 + 5y + y2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 5y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + 5y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 5y + y2 = 0 + 7
5y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
5y + y2 = 7

The y term is 5y.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5y + 6.25 + y2 = 7 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 7 + 6.25

Combine like terms: 7 + 6.25 = 13.25
6.25 + 5y + y2 = 13.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(y + 2.5) equal to 3.640054945 and -3.640054945.


Subproblem 1
y + 2.5 = 3.640054945

Simplifying
y + 2.5 = 3.640054945

Reorder the terms:
2.5 + y = 3.640054945

Solving
2.5 + y = 3.640054945

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = 3.640054945 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 3.640054945 + -2.5
y = 3.640054945 + -2.5

Combine like terms: 3.640054945 + -2.5 = 1.140054945
y = 1.140054945

Simplifying
y = 1.140054945


Subproblem 2
y + 2.5 = -3.640054945

Simplifying
y + 2.5 = -3.640054945

Reorder the terms:
2.5 + y = -3.640054945

Solving
2.5 + y = -3.640054945

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = -3.640054945 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -3.640054945 + -2.5
y = -3.640054945 + -2.5

Combine like terms: -3.640054945 + -2.5 = -6.140054945
y = -6.140054945

Simplifying
y = -6.140054945


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.140054945, -6.140054945}

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