10y^2-30y+13=0

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Solution for 10y^2-30y+13=0 equation:


Simplifying
10y2 + -30y + 13 = 0

Reorder the terms:
13 + -30y + 10y2 = 0

Solving
13 + -30y + 10y2 = 0

Solving for variable 'y'.

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

Divide each side by '10'.
1.3 + -3y + y2 = 0

Move the constant term to the right:

Add '-1.3' to each side of the equation.
1.3 + -3y + -1.3 + y2 = 0 + -1.3

Reorder the terms:
1.3 + -1.3 + -3y + y2 = 0 + -1.3

Combine like terms: 1.3 + -1.3 = 0.0
0.0 + -3y + y2 = 0 + -1.3
-3y + y2 = 0 + -1.3

Combine like terms: 0 + -1.3 = -1.3
-3y + y2 = -1.3

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 = -1.3 + 2.25

Reorder the terms:
2.25 + -3y + y2 = -1.3 + 2.25

Combine like terms: -1.3 + 2.25 = 0.95
2.25 + -3y + y2 = 0.95

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 0.95

Calculate the square root of the right side: 0.974679434

Break this problem into two subproblems by setting 
(y + -1.5) equal to 0.974679434 and -0.974679434.

Subproblem 1

y + -1.5 = 0.974679434 Simplifying y + -1.5 = 0.974679434 Reorder the terms: -1.5 + y = 0.974679434 Solving -1.5 + y = 0.974679434 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 = 0.974679434 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 0.974679434 + 1.5 y = 0.974679434 + 1.5 Combine like terms: 0.974679434 + 1.5 = 2.474679434 y = 2.474679434 Simplifying y = 2.474679434

Subproblem 2

y + -1.5 = -0.974679434 Simplifying y + -1.5 = -0.974679434 Reorder the terms: -1.5 + y = -0.974679434 Solving -1.5 + y = -0.974679434 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 = -0.974679434 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -0.974679434 + 1.5 y = -0.974679434 + 1.5 Combine like terms: -0.974679434 + 1.5 = 0.525320566 y = 0.525320566 Simplifying y = 0.525320566

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.474679434, 0.525320566}

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