-y^2+4y=1

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Solution for -y^2+4y=1 equation:


Simplifying
-1y2 + 4y = 1

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

Solving
4y + -1y2 = 1

Solving for variable 'y'.

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

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

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

Divide each side by '-1'.
1 + -4y + y2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + -4y + y2 = 0 + -1

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

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

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 = -1 + 4

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

Combine like terms: -1 + 4 = 3
4 + -4y + y2 = 3

Factor a perfect square on the left side:
(y + -2)(y + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(y + -2) equal to 1.732050808 and -1.732050808.

Subproblem 1

y + -2 = 1.732050808 Simplifying y + -2 = 1.732050808 Reorder the terms: -2 + y = 1.732050808 Solving -2 + y = 1.732050808 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 = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + y = 1.732050808 + 2 y = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 y = 3.732050808 Simplifying y = 3.732050808

Subproblem 2

y + -2 = -1.732050808 Simplifying y + -2 = -1.732050808 Reorder the terms: -2 + y = -1.732050808 Solving -2 + y = -1.732050808 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 = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + y = -1.732050808 + 2 y = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 y = 0.267949192 Simplifying y = 0.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.732050808, 0.267949192}

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