2y^2-69y+13=0

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Solution for 2y^2-69y+13=0 equation:


Simplifying
2y2 + -69y + 13 = 0

Reorder the terms:
13 + -69y + 2y2 = 0

Solving
13 + -69y + 2y2 = 0

Solving for variable 'y'.

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

Divide each side by '2'.
6.5 + -34.5y + y2 = 0

Move the constant term to the right:

Add '-6.5' to each side of the equation.
6.5 + -34.5y + -6.5 + y2 = 0 + -6.5

Reorder the terms:
6.5 + -6.5 + -34.5y + y2 = 0 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + -34.5y + y2 = 0 + -6.5
-34.5y + y2 = 0 + -6.5

Combine like terms: 0 + -6.5 = -6.5
-34.5y + y2 = -6.5

The y term is -34.5y.  Take half its coefficient (-17.25).
Square it (297.5625) and add it to both sides.

Add '297.5625' to each side of the equation.
-34.5y + 297.5625 + y2 = -6.5 + 297.5625

Reorder the terms:
297.5625 + -34.5y + y2 = -6.5 + 297.5625

Combine like terms: -6.5 + 297.5625 = 291.0625
297.5625 + -34.5y + y2 = 291.0625

Factor a perfect square on the left side:
(y + -17.25)(y + -17.25) = 291.0625

Calculate the square root of the right side: 17.060553918

Break this problem into two subproblems by setting 
(y + -17.25) equal to 17.060553918 and -17.060553918.

Subproblem 1

y + -17.25 = 17.060553918 Simplifying y + -17.25 = 17.060553918 Reorder the terms: -17.25 + y = 17.060553918 Solving -17.25 + y = 17.060553918 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '17.25' to each side of the equation. -17.25 + 17.25 + y = 17.060553918 + 17.25 Combine like terms: -17.25 + 17.25 = 0.00 0.00 + y = 17.060553918 + 17.25 y = 17.060553918 + 17.25 Combine like terms: 17.060553918 + 17.25 = 34.310553918 y = 34.310553918 Simplifying y = 34.310553918

Subproblem 2

y + -17.25 = -17.060553918 Simplifying y + -17.25 = -17.060553918 Reorder the terms: -17.25 + y = -17.060553918 Solving -17.25 + y = -17.060553918 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '17.25' to each side of the equation. -17.25 + 17.25 + y = -17.060553918 + 17.25 Combine like terms: -17.25 + 17.25 = 0.00 0.00 + y = -17.060553918 + 17.25 y = -17.060553918 + 17.25 Combine like terms: -17.060553918 + 17.25 = 0.189446082 y = 0.189446082 Simplifying y = 0.189446082

Solution

The solution to the problem is based on the solutions from the subproblems. y = {34.310553918, 0.189446082}

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