4y+y^2=10

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Solution for 4y+y^2=10 equation: 


Simplifying
4y + y2 = 10

Solving
4y + y2 = 10

Solving for variable 'y'.

Reorder the terms:
-10 + 4y + y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 4y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 4y + 10 + y2 = 0 + 10

Reorder the terms:
-10 + 10 + 4y + y2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 4y + y2 = 0 + 10
4y + y2 = 0 + 10

Combine like terms: 0 + 10 = 10
4y + y2 = 10

The y term is 4y.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4y + 4 + y2 = 10 + 4

Reorder the terms:
4 + 4y + y2 = 10 + 4

Combine like terms: 10 + 4 = 14
4 + 4y + y2 = 14

Factor a perfect square on the left side:
(y + 2)(y + 2) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(y + 2) equal to 3.741657387 and -3.741657387.


Subproblem 1
y + 2 = 3.741657387

Simplifying
y + 2 = 3.741657387

Reorder the terms:
2 + y = 3.741657387

Solving
2 + y = 3.741657387

Solving for variable 'y'.

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

Add '-2' to each side of the equation.
2 + -2 + y = 3.741657387 + -2

Combine like terms: 2 + -2 = 0
0 + y = 3.741657387 + -2
y = 3.741657387 + -2

Combine like terms: 3.741657387 + -2 = 1.741657387
y = 1.741657387

Simplifying
y = 1.741657387


Subproblem 2
y + 2 = -3.741657387

Simplifying
y + 2 = -3.741657387

Reorder the terms:
2 + y = -3.741657387

Solving
2 + y = -3.741657387

Solving for variable 'y'.

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

Add '-2' to each side of the equation.
2 + -2 + y = -3.741657387 + -2

Combine like terms: 2 + -2 = 0
0 + y = -3.741657387 + -2
y = -3.741657387 + -2

Combine like terms: -3.741657387 + -2 = -5.741657387
y = -5.741657387

Simplifying
y = -5.741657387


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.741657387, -5.741657387}

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