(y^2+4)=2y^2-y-10

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Solution for (y^2+4)=2y^2-y-10 equation:


Simplifying
(y2 + 4) = 2y2 + -1y + -10

Reorder the terms:
(4 + y2) = 2y2 + -1y + -10

Remove parenthesis around (4 + y2)
4 + y2 = 2y2 + -1y + -10

Reorder the terms:
4 + y2 = -10 + -1y + 2y2

Solving
4 + y2 = -10 + -1y + 2y2

Solving for variable 'y'.

Reorder the terms:
4 + 10 + y + y2 + -2y2 = -10 + -1y + 2y2 + 10 + y + -2y2

Combine like terms: 4 + 10 = 14
14 + y + y2 + -2y2 = -10 + -1y + 2y2 + 10 + y + -2y2

Combine like terms: y2 + -2y2 = -1y2
14 + y + -1y2 = -10 + -1y + 2y2 + 10 + y + -2y2

Reorder the terms:
14 + y + -1y2 = -10 + 10 + -1y + y + 2y2 + -2y2

Combine like terms: -10 + 10 = 0
14 + y + -1y2 = 0 + -1y + y + 2y2 + -2y2
14 + y + -1y2 = -1y + y + 2y2 + -2y2

Combine like terms: -1y + y = 0
14 + y + -1y2 = 0 + 2y2 + -2y2
14 + y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
14 + y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-14 + -1y + y2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + -1y + 14 + y2 = 0 + 14

Reorder the terms:
-14 + 14 + -1y + y2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + -1y + y2 = 0 + 14
-1y + y2 = 0 + 14

Combine like terms: 0 + 14 = 14
-1y + y2 = 14

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1y + 0.25 + y2 = 14 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 14 + 0.25

Combine like terms: 14 + 0.25 = 14.25
0.25 + -1y + y2 = 14.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(y + 0.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

y + 0.5 = 3.774917218 Simplifying y + 0.5 = 3.774917218 Reorder the terms: 0.5 + y = 3.774917218 Solving 0.5 + y = 3.774917218 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 3.774917218 + -0.5 y = 3.774917218 + -0.5 Combine like terms: 3.774917218 + -0.5 = 3.274917218 y = 3.274917218 Simplifying y = 3.274917218

Subproblem 2

y + 0.5 = -3.774917218 Simplifying y + 0.5 = -3.774917218 Reorder the terms: 0.5 + y = -3.774917218 Solving 0.5 + y = -3.774917218 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -3.774917218 + -0.5 y = -3.774917218 + -0.5 Combine like terms: -3.774917218 + -0.5 = -4.274917218 y = -4.274917218 Simplifying y = -4.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.274917218, -4.274917218}

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