18y+7y^2=180

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Solution for 18y+7y^2=180 equation:


Simplifying
18y + 7y2 = 180

Solving
18y + 7y2 = 180

Solving for variable 'y'.

Reorder the terms:
-180 + 18y + 7y2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 18y + 7y2 = 0

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

Divide each side by '7'.
-25.71428571 + 2.571428571y + y2 = 0

Move the constant term to the right:

Add '25.71428571' to each side of the equation.
-25.71428571 + 2.571428571y + 25.71428571 + y2 = 0 + 25.71428571

Reorder the terms:
-25.71428571 + 25.71428571 + 2.571428571y + y2 = 0 + 25.71428571

Combine like terms: -25.71428571 + 25.71428571 = 0.00000000
0.00000000 + 2.571428571y + y2 = 0 + 25.71428571
2.571428571y + y2 = 0 + 25.71428571

Combine like terms: 0 + 25.71428571 = 25.71428571
2.571428571y + y2 = 25.71428571

The y term is 2.571428571y.  Take half its coefficient (1.285714286).
Square it (1.653061225) and add it to both sides.

Add '1.653061225' to each side of the equation.
2.571428571y + 1.653061225 + y2 = 25.71428571 + 1.653061225

Reorder the terms:
1.653061225 + 2.571428571y + y2 = 25.71428571 + 1.653061225

Combine like terms: 25.71428571 + 1.653061225 = 27.367346935
1.653061225 + 2.571428571y + y2 = 27.367346935

Factor a perfect square on the left side:
(y + 1.285714286)(y + 1.285714286) = 27.367346935

Calculate the square root of the right side: 5.231380978

Break this problem into two subproblems by setting 
(y + 1.285714286) equal to 5.231380978 and -5.231380978.

Subproblem 1

y + 1.285714286 = 5.231380978 Simplifying y + 1.285714286 = 5.231380978 Reorder the terms: 1.285714286 + y = 5.231380978 Solving 1.285714286 + y = 5.231380978 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.285714286' to each side of the equation. 1.285714286 + -1.285714286 + y = 5.231380978 + -1.285714286 Combine like terms: 1.285714286 + -1.285714286 = 0.000000000 0.000000000 + y = 5.231380978 + -1.285714286 y = 5.231380978 + -1.285714286 Combine like terms: 5.231380978 + -1.285714286 = 3.945666692 y = 3.945666692 Simplifying y = 3.945666692

Subproblem 2

y + 1.285714286 = -5.231380978 Simplifying y + 1.285714286 = -5.231380978 Reorder the terms: 1.285714286 + y = -5.231380978 Solving 1.285714286 + y = -5.231380978 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.285714286' to each side of the equation. 1.285714286 + -1.285714286 + y = -5.231380978 + -1.285714286 Combine like terms: 1.285714286 + -1.285714286 = 0.000000000 0.000000000 + y = -5.231380978 + -1.285714286 y = -5.231380978 + -1.285714286 Combine like terms: -5.231380978 + -1.285714286 = -6.517095264 y = -6.517095264 Simplifying y = -6.517095264

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.945666692, -6.517095264}

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