(-y^2+2y-9)-(-5y^2-2)=0

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Solution for (-y^2+2y-9)-(-5y^2-2)=0 equation:


Simplifying
(-1y2 + 2y + -9) + -1(-5y2 + -2) = 0

Reorder the terms:
(-9 + 2y + -1y2) + -1(-5y2 + -2) = 0

Remove parenthesis around (-9 + 2y + -1y2)
-9 + 2y + -1y2 + -1(-5y2 + -2) = 0

Reorder the terms:
-9 + 2y + -1y2 + -1(-2 + -5y2) = 0
-9 + 2y + -1y2 + (-2 * -1 + -5y2 * -1) = 0
-9 + 2y + -1y2 + (2 + 5y2) = 0

Reorder the terms:
-9 + 2 + 2y + -1y2 + 5y2 = 0

Combine like terms: -9 + 2 = -7
-7 + 2y + -1y2 + 5y2 = 0

Combine like terms: -1y2 + 5y2 = 4y2
-7 + 2y + 4y2 = 0

Solving
-7 + 2y + 4y2 = 0

Solving for variable 'y'.

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

Divide each side by '4'.
-1.75 + 0.5y + y2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + 0.5y + 1.75 + y2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + 0.5y + y2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + 0.5y + y2 = 0 + 1.75
0.5y + y2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
0.5y + y2 = 1.75

The y term is 0.5y.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5y + 0.0625 + y2 = 1.75 + 0.0625

Reorder the terms:
0.0625 + 0.5y + y2 = 1.75 + 0.0625

Combine like terms: 1.75 + 0.0625 = 1.8125
0.0625 + 0.5y + y2 = 1.8125

Factor a perfect square on the left side:
(y + 0.25)(y + 0.25) = 1.8125

Calculate the square root of the right side: 1.346291202

Break this problem into two subproblems by setting 
(y + 0.25) equal to 1.346291202 and -1.346291202.

Subproblem 1

y + 0.25 = 1.346291202 Simplifying y + 0.25 = 1.346291202 Reorder the terms: 0.25 + y = 1.346291202 Solving 0.25 + y = 1.346291202 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = 1.346291202 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = 1.346291202 + -0.25 y = 1.346291202 + -0.25 Combine like terms: 1.346291202 + -0.25 = 1.096291202 y = 1.096291202 Simplifying y = 1.096291202

Subproblem 2

y + 0.25 = -1.346291202 Simplifying y + 0.25 = -1.346291202 Reorder the terms: 0.25 + y = -1.346291202 Solving 0.25 + y = -1.346291202 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = -1.346291202 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = -1.346291202 + -0.25 y = -1.346291202 + -0.25 Combine like terms: -1.346291202 + -0.25 = -1.596291202 y = -1.596291202 Simplifying y = -1.596291202

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.096291202, -1.596291202}

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