20y^2-10y-9=0

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Solution for 20y^2-10y-9=0 equation:


Simplifying
20y2 + -10y + -9 = 0

Reorder the terms:
-9 + -10y + 20y2 = 0

Solving
-9 + -10y + 20y2 = 0

Solving for variable 'y'.

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

Divide each side by '20'.
-0.45 + -0.5y + y2 = 0

Move the constant term to the right:

Add '0.45' to each side of the equation.
-0.45 + -0.5y + 0.45 + y2 = 0 + 0.45

Reorder the terms:
-0.45 + 0.45 + -0.5y + y2 = 0 + 0.45

Combine like terms: -0.45 + 0.45 = 0.00
0.00 + -0.5y + y2 = 0 + 0.45
-0.5y + y2 = 0 + 0.45

Combine like terms: 0 + 0.45 = 0.45
-0.5y + y2 = 0.45

The y term is -0.5y.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5y + 0.0625 + y2 = 0.45 + 0.0625

Reorder the terms:
0.0625 + -0.5y + y2 = 0.45 + 0.0625

Combine like terms: 0.45 + 0.0625 = 0.5125
0.0625 + -0.5y + y2 = 0.5125

Factor a perfect square on the left side:
(y + -0.25)(y + -0.25) = 0.5125

Calculate the square root of the right side: 0.715891053

Break this problem into two subproblems by setting 
(y + -0.25) equal to 0.715891053 and -0.715891053.

Subproblem 1

y + -0.25 = 0.715891053 Simplifying y + -0.25 = 0.715891053 Reorder the terms: -0.25 + y = 0.715891053 Solving -0.25 + y = 0.715891053 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = 0.715891053 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = 0.715891053 + 0.25 y = 0.715891053 + 0.25 Combine like terms: 0.715891053 + 0.25 = 0.965891053 y = 0.965891053 Simplifying y = 0.965891053

Subproblem 2

y + -0.25 = -0.715891053 Simplifying y + -0.25 = -0.715891053 Reorder the terms: -0.25 + y = -0.715891053 Solving -0.25 + y = -0.715891053 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = -0.715891053 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = -0.715891053 + 0.25 y = -0.715891053 + 0.25 Combine like terms: -0.715891053 + 0.25 = -0.465891053 y = -0.465891053 Simplifying y = -0.465891053

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.965891053, -0.465891053}

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