y^2=13y-6

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Solution for y^2=13y-6 equation:


Simplifying
y2 = 13y + -6

Reorder the terms:
y2 = -6 + 13y

Solving
y2 = -6 + 13y

Solving for variable 'y'.

Reorder the terms:
6 + -13y + y2 = -6 + 13y + 6 + -13y

Reorder the terms:
6 + -13y + y2 = -6 + 6 + 13y + -13y

Combine like terms: -6 + 6 = 0
6 + -13y + y2 = 0 + 13y + -13y
6 + -13y + y2 = 13y + -13y

Combine like terms: 13y + -13y = 0
6 + -13y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -13y + -6 + y2 = 0 + -6

Reorder the terms:
6 + -6 + -13y + y2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -13y + y2 = 0 + -6
-13y + y2 = 0 + -6

Combine like terms: 0 + -6 = -6
-13y + y2 = -6

The y term is -13y.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13y + 42.25 + y2 = -6 + 42.25

Reorder the terms:
42.25 + -13y + y2 = -6 + 42.25

Combine like terms: -6 + 42.25 = 36.25
42.25 + -13y + y2 = 36.25

Factor a perfect square on the left side:
(y + -6.5)(y + -6.5) = 36.25

Calculate the square root of the right side: 6.020797289

Break this problem into two subproblems by setting 
(y + -6.5) equal to 6.020797289 and -6.020797289.

Subproblem 1

y + -6.5 = 6.020797289 Simplifying y + -6.5 = 6.020797289 Reorder the terms: -6.5 + y = 6.020797289 Solving -6.5 + y = 6.020797289 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + y = 6.020797289 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 6.020797289 + 6.5 y = 6.020797289 + 6.5 Combine like terms: 6.020797289 + 6.5 = 12.520797289 y = 12.520797289 Simplifying y = 12.520797289

Subproblem 2

y + -6.5 = -6.020797289 Simplifying y + -6.5 = -6.020797289 Reorder the terms: -6.5 + y = -6.020797289 Solving -6.5 + y = -6.020797289 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + y = -6.020797289 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -6.020797289 + 6.5 y = -6.020797289 + 6.5 Combine like terms: -6.020797289 + 6.5 = 0.479202711 y = 0.479202711 Simplifying y = 0.479202711

Solution

The solution to the problem is based on the solutions from the subproblems. y = {12.520797289, 0.479202711}

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