(2y^2)+(3y)=7

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Solution for (2y^2)+(3y)=7 equation:


Simplifying
(2y2) + (3y) = 7

Reorder the terms:
(3y) + (2y2) = 7

Solving
(3y) + (2y2) = 7

Solving for variable 'y'.

Reorder the terms:
-7 + (3y) + (2y2) = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + (3y) + (2y2) = 0

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

Divide each side by '2'.
-3.5 + (1.5y) + y2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + (1.5y) + 3.5 + y2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + (1.5y) + y2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + (1.5y) + y2 = 0 + 3.5
(1.5y) + y2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
(1.5y) + y2 = 3.5

The y term is (1.5y).  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
(1.5y) + 0.5625 + y2 = 3.5 + 0.5625

Reorder the terms:
0.5625 + (1.5y) + y2 = 3.5 + 0.5625

Combine like terms: 3.5 + 0.5625 = 4.0625
0.5625 + (1.5y) + y2 = 4.0625

Factor a perfect square on the left side:
((y) + 0.75)((y) + 0.75) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
((y) + 0.75) equal to 2.015564437 and -2.015564437.

Subproblem 1

(y) + 0.75 = 2.015564437 Simplifying (y) + 0.75 = 2.015564437 y + 0.75 = 2.015564437 Reorder the terms: 0.75 + y = 2.015564437 Solving 0.75 + y = 2.015564437 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 2.015564437 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 2.015564437 + -0.75 y = 2.015564437 + -0.75 Combine like terms: 2.015564437 + -0.75 = 1.265564437 y = 1.265564437 Simplifying y = 1.265564437

Subproblem 2

(y) + 0.75 = -2.015564437 Simplifying (y) + 0.75 = -2.015564437 y + 0.75 = -2.015564437 Reorder the terms: 0.75 + y = -2.015564437 Solving 0.75 + y = -2.015564437 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -2.015564437 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -2.015564437 + -0.75 y = -2.015564437 + -0.75 Combine like terms: -2.015564437 + -0.75 = -2.765564437 y = -2.765564437 Simplifying y = -2.765564437

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.265564437, -2.765564437}

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