y^2-10y=1

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Solution for y^2-10y=1 equation:


Simplifying
y2 + -10y = 1

Reorder the terms:
-10y + y2 = 1

Solving
-10y + y2 = 1

Solving for variable 'y'.

Reorder the terms:
-1 + -10y + y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -10y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -10y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -10y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -10y + y2 = 0 + 1
-10y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-10y + y2 = 1

The y term is -10y.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10y + 25 + y2 = 1 + 25

Reorder the terms:
25 + -10y + y2 = 1 + 25

Combine like terms: 1 + 25 = 26
25 + -10y + y2 = 26

Factor a perfect square on the left side:
(y + -5)(y + -5) = 26

Calculate the square root of the right side: 5.099019514

Break this problem into two subproblems by setting 
(y + -5) equal to 5.099019514 and -5.099019514.

Subproblem 1

y + -5 = 5.099019514 Simplifying y + -5 = 5.099019514 Reorder the terms: -5 + y = 5.099019514 Solving -5 + y = 5.099019514 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = 5.099019514 + 5 Combine like terms: -5 + 5 = 0 0 + y = 5.099019514 + 5 y = 5.099019514 + 5 Combine like terms: 5.099019514 + 5 = 10.099019514 y = 10.099019514 Simplifying y = 10.099019514

Subproblem 2

y + -5 = -5.099019514 Simplifying y + -5 = -5.099019514 Reorder the terms: -5 + y = -5.099019514 Solving -5 + y = -5.099019514 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = -5.099019514 + 5 Combine like terms: -5 + 5 = 0 0 + y = -5.099019514 + 5 y = -5.099019514 + 5 Combine like terms: -5.099019514 + 5 = -0.099019514 y = -0.099019514 Simplifying y = -0.099019514

Solution

The solution to the problem is based on the solutions from the subproblems. y = {10.099019514, -0.099019514}

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