y^2+y+1=14

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Solution for y^2+y+1=14 equation:


Simplifying
y2 + y + 1 = 14

Reorder the terms:
1 + y + y2 = 14

Solving
1 + y + y2 = 14

Solving for variable 'y'.

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

Combine like terms: 1 + -14 = -13
-13 + y + y2 = 14 + -14

Combine like terms: 14 + -14 = 0
-13 + y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + y + 13 + y2 = 0 + 13

Reorder the terms:
-13 + 13 + y + y2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + y + y2 = 0 + 13
y + y2 = 0 + 13

Combine like terms: 0 + 13 = 13
y + y2 = 13

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.
y + 0.25 + y2 = 13 + 0.25

Reorder the terms:
0.25 + y + y2 = 13 + 0.25

Combine like terms: 13 + 0.25 = 13.25
0.25 + y + y2 = 13.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(y + 0.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

y + 0.5 = 3.640054945 Simplifying y + 0.5 = 3.640054945 Reorder the terms: 0.5 + y = 3.640054945 Solving 0.5 + y = 3.640054945 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 = 3.640054945 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 3.640054945 + -0.5 y = 3.640054945 + -0.5 Combine like terms: 3.640054945 + -0.5 = 3.140054945 y = 3.140054945 Simplifying y = 3.140054945

Subproblem 2

y + 0.5 = -3.640054945 Simplifying y + 0.5 = -3.640054945 Reorder the terms: 0.5 + y = -3.640054945 Solving 0.5 + y = -3.640054945 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 = -3.640054945 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -3.640054945 + -0.5 y = -3.640054945 + -0.5 Combine like terms: -3.640054945 + -0.5 = -4.140054945 y = -4.140054945 Simplifying y = -4.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.140054945, -4.140054945}

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