y=y^2-5

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Solution for y=y^2-5 equation:


Simplifying
y = y2 + -5

Reorder the terms:
y = -5 + y2

Solving
y = -5 + y2

Solving for variable 'y'.

Reorder the terms:
5 + y + -1y2 = -5 + y2 + 5 + -1y2

Reorder the terms:
5 + y + -1y2 = -5 + 5 + y2 + -1y2

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

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

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

Divide each side by '-1'.
-5 + -1y + y2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1y + 5 + y2 = 0 + 5

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

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

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

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 = 5 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1y + y2 = 5.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(y + 0.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

y + 0.5 = 2.291287847 Simplifying y + 0.5 = 2.291287847 Reorder the terms: 0.5 + y = 2.291287847 Solving 0.5 + y = 2.291287847 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 = 2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 2.291287847 + -0.5 y = 2.291287847 + -0.5 Combine like terms: 2.291287847 + -0.5 = 1.791287847 y = 1.791287847 Simplifying y = 1.791287847

Subproblem 2

y + 0.5 = -2.291287847 Simplifying y + 0.5 = -2.291287847 Reorder the terms: 0.5 + y = -2.291287847 Solving 0.5 + y = -2.291287847 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 = -2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -2.291287847 + -0.5 y = -2.291287847 + -0.5 Combine like terms: -2.291287847 + -0.5 = -2.791287847 y = -2.791287847 Simplifying y = -2.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.791287847, -2.791287847}

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