(y-5)(y-5)=2y^2+5y+11

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Solution for (y-5)(y-5)=2y^2+5y+11 equation:


Simplifying
(y + -5)(y + -5) = 2y2 + 5y + 11

Reorder the terms:
(-5 + y)(y + -5) = 2y2 + 5y + 11

Reorder the terms:
(-5 + y)(-5 + y) = 2y2 + 5y + 11

Multiply (-5 + y) * (-5 + y)
(-5(-5 + y) + y(-5 + y)) = 2y2 + 5y + 11
((-5 * -5 + y * -5) + y(-5 + y)) = 2y2 + 5y + 11
((25 + -5y) + y(-5 + y)) = 2y2 + 5y + 11
(25 + -5y + (-5 * y + y * y)) = 2y2 + 5y + 11
(25 + -5y + (-5y + y2)) = 2y2 + 5y + 11

Combine like terms: -5y + -5y = -10y
(25 + -10y + y2) = 2y2 + 5y + 11

Reorder the terms:
25 + -10y + y2 = 11 + 5y + 2y2

Solving
25 + -10y + y2 = 11 + 5y + 2y2

Solving for variable 'y'.

Reorder the terms:
25 + -11 + -10y + -5y + y2 + -2y2 = 11 + 5y + 2y2 + -11 + -5y + -2y2

Combine like terms: 25 + -11 = 14
14 + -10y + -5y + y2 + -2y2 = 11 + 5y + 2y2 + -11 + -5y + -2y2

Combine like terms: -10y + -5y = -15y
14 + -15y + y2 + -2y2 = 11 + 5y + 2y2 + -11 + -5y + -2y2

Combine like terms: y2 + -2y2 = -1y2
14 + -15y + -1y2 = 11 + 5y + 2y2 + -11 + -5y + -2y2

Reorder the terms:
14 + -15y + -1y2 = 11 + -11 + 5y + -5y + 2y2 + -2y2

Combine like terms: 11 + -11 = 0
14 + -15y + -1y2 = 0 + 5y + -5y + 2y2 + -2y2
14 + -15y + -1y2 = 5y + -5y + 2y2 + -2y2

Combine like terms: 5y + -5y = 0
14 + -15y + -1y2 = 0 + 2y2 + -2y2
14 + -15y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
14 + -15y + -1y2 = 0

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

Divide each side by '-1'.
-14 + 15y + y2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 15y + 14 + y2 = 0 + 14

Reorder the terms:
-14 + 14 + 15y + y2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 15y + y2 = 0 + 14
15y + y2 = 0 + 14

Combine like terms: 0 + 14 = 14
15y + y2 = 14

The y term is 15y.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15y + 56.25 + y2 = 14 + 56.25

Reorder the terms:
56.25 + 15y + y2 = 14 + 56.25

Combine like terms: 14 + 56.25 = 70.25
56.25 + 15y + y2 = 70.25

Factor a perfect square on the left side:
(y + 7.5)(y + 7.5) = 70.25

Calculate the square root of the right side: 8.381527307

Break this problem into two subproblems by setting 
(y + 7.5) equal to 8.381527307 and -8.381527307.

Subproblem 1

y + 7.5 = 8.381527307 Simplifying y + 7.5 = 8.381527307 Reorder the terms: 7.5 + y = 8.381527307 Solving 7.5 + y = 8.381527307 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + y = 8.381527307 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + y = 8.381527307 + -7.5 y = 8.381527307 + -7.5 Combine like terms: 8.381527307 + -7.5 = 0.881527307 y = 0.881527307 Simplifying y = 0.881527307

Subproblem 2

y + 7.5 = -8.381527307 Simplifying y + 7.5 = -8.381527307 Reorder the terms: 7.5 + y = -8.381527307 Solving 7.5 + y = -8.381527307 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + y = -8.381527307 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + y = -8.381527307 + -7.5 y = -8.381527307 + -7.5 Combine like terms: -8.381527307 + -7.5 = -15.881527307 y = -15.881527307 Simplifying y = -15.881527307

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.881527307, -15.881527307}

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