y^2=13y+1

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Solution for y^2=13y+1 equation:


Simplifying
y2 = 13y + 1

Reorder the terms:
y2 = 1 + 13y

Solving
y2 = 1 + 13y

Solving for variable 'y'.

Reorder the terms:
-1 + -13y + y2 = 1 + 13y + -1 + -13y

Reorder the terms:
-1 + -13y + y2 = 1 + -1 + 13y + -13y

Combine like terms: 1 + -1 = 0
-1 + -13y + y2 = 0 + 13y + -13y
-1 + -13y + y2 = 13y + -13y

Combine like terms: 13y + -13y = 0
-1 + -13y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -13y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -13y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -13y + y2 = 0 + 1
-13y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-13y + y2 = 1

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 = 1 + 42.25

Reorder the terms:
42.25 + -13y + y2 = 1 + 42.25

Combine like terms: 1 + 42.25 = 43.25
42.25 + -13y + y2 = 43.25

Factor a perfect square on the left side:
(y + -6.5)(y + -6.5) = 43.25

Calculate the square root of the right side: 6.576473219

Break this problem into two subproblems by setting 
(y + -6.5) equal to 6.576473219 and -6.576473219.

Subproblem 1

y + -6.5 = 6.576473219 Simplifying y + -6.5 = 6.576473219 Reorder the terms: -6.5 + y = 6.576473219 Solving -6.5 + y = 6.576473219 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 = 6.576473219 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 6.576473219 + 6.5 y = 6.576473219 + 6.5 Combine like terms: 6.576473219 + 6.5 = 13.076473219 y = 13.076473219 Simplifying y = 13.076473219

Subproblem 2

y + -6.5 = -6.576473219 Simplifying y + -6.5 = -6.576473219 Reorder the terms: -6.5 + y = -6.576473219 Solving -6.5 + y = -6.576473219 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 = -6.576473219 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -6.576473219 + 6.5 y = -6.576473219 + 6.5 Combine like terms: -6.576473219 + 6.5 = -0.076473219 y = -0.076473219 Simplifying y = -0.076473219

Solution

The solution to the problem is based on the solutions from the subproblems. y = {13.076473219, -0.076473219}

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