y^2=3y+1

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Solution for y^2=3y+1 equation:


Simplifying
y2 = 3y + 1

Reorder the terms:
y2 = 1 + 3y

Solving
y2 = 1 + 3y

Solving for variable 'y'.

Reorder the terms:
-1 + -3y + y2 = 1 + 3y + -1 + -3y

Reorder the terms:
-1 + -3y + y2 = 1 + -1 + 3y + -3y

Combine like terms: 1 + -1 = 0
-1 + -3y + y2 = 0 + 3y + -3y
-1 + -3y + y2 = 3y + -3y

Combine like terms: 3y + -3y = 0
-1 + -3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -3y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3y + y2 = 0 + 1
-3y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3y + y2 = 1

The y term is -3y.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3y + 2.25 + y2 = 1 + 2.25

Reorder the terms:
2.25 + -3y + y2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + -3y + y2 = 3.25

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(y + -1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

y + -1.5 = 1.802775638 Simplifying y + -1.5 = 1.802775638 Reorder the terms: -1.5 + y = 1.802775638 Solving -1.5 + y = 1.802775638 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = 1.802775638 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 1.802775638 + 1.5 y = 1.802775638 + 1.5 Combine like terms: 1.802775638 + 1.5 = 3.302775638 y = 3.302775638 Simplifying y = 3.302775638

Subproblem 2

y + -1.5 = -1.802775638 Simplifying y + -1.5 = -1.802775638 Reorder the terms: -1.5 + y = -1.802775638 Solving -1.5 + y = -1.802775638 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = -1.802775638 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -1.802775638 + 1.5 y = -1.802775638 + 1.5 Combine like terms: -1.802775638 + 1.5 = -0.302775638 y = -0.302775638 Simplifying y = -0.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.302775638, -0.302775638}

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