20y^2+5y=9

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Solution for 20y^2+5y=9 equation:


Simplifying
20y2 + 5y = 9

Reorder the terms:
5y + 20y2 = 9

Solving
5y + 20y2 = 9

Solving for variable 'y'.

Reorder the terms:
-9 + 5y + 20y2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 5y + 20y2 = 0

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

Divide each side by '20'.
-0.45 + 0.25y + y2 = 0

Move the constant term to the right:

Add '0.45' to each side of the equation.
-0.45 + 0.25y + 0.45 + y2 = 0 + 0.45

Reorder the terms:
-0.45 + 0.45 + 0.25y + y2 = 0 + 0.45

Combine like terms: -0.45 + 0.45 = 0.00
0.00 + 0.25y + y2 = 0 + 0.45
0.25y + y2 = 0 + 0.45

Combine like terms: 0 + 0.45 = 0.45
0.25y + y2 = 0.45

The y term is 0.25y.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25y + 0.015625 + y2 = 0.45 + 0.015625

Reorder the terms:
0.015625 + 0.25y + y2 = 0.45 + 0.015625

Combine like terms: 0.45 + 0.015625 = 0.465625
0.015625 + 0.25y + y2 = 0.465625

Factor a perfect square on the left side:
(y + 0.125)(y + 0.125) = 0.465625

Calculate the square root of the right side: 0.682367203

Break this problem into two subproblems by setting 
(y + 0.125) equal to 0.682367203 and -0.682367203.

Subproblem 1

y + 0.125 = 0.682367203 Simplifying y + 0.125 = 0.682367203 Reorder the terms: 0.125 + y = 0.682367203 Solving 0.125 + y = 0.682367203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + y = 0.682367203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + y = 0.682367203 + -0.125 y = 0.682367203 + -0.125 Combine like terms: 0.682367203 + -0.125 = 0.557367203 y = 0.557367203 Simplifying y = 0.557367203

Subproblem 2

y + 0.125 = -0.682367203 Simplifying y + 0.125 = -0.682367203 Reorder the terms: 0.125 + y = -0.682367203 Solving 0.125 + y = -0.682367203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + y = -0.682367203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + y = -0.682367203 + -0.125 y = -0.682367203 + -0.125 Combine like terms: -0.682367203 + -0.125 = -0.807367203 y = -0.807367203 Simplifying y = -0.807367203

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.557367203, -0.807367203}

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