2y^2+15y+21=0

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Solution for 2y^2+15y+21=0 equation:


Simplifying
2y2 + 15y + 21 = 0

Reorder the terms:
21 + 15y + 2y2 = 0

Solving
21 + 15y + 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'.
10.5 + 7.5y + y2 = 0

Move the constant term to the right:

Add '-10.5' to each side of the equation.
10.5 + 7.5y + -10.5 + y2 = 0 + -10.5

Reorder the terms:
10.5 + -10.5 + 7.5y + y2 = 0 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + 7.5y + y2 = 0 + -10.5
7.5y + y2 = 0 + -10.5

Combine like terms: 0 + -10.5 = -10.5
7.5y + y2 = -10.5

The y term is 7.5y.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5y + 14.0625 + y2 = -10.5 + 14.0625

Reorder the terms:
14.0625 + 7.5y + y2 = -10.5 + 14.0625

Combine like terms: -10.5 + 14.0625 = 3.5625
14.0625 + 7.5y + y2 = 3.5625

Factor a perfect square on the left side:
(y + 3.75)(y + 3.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(y + 3.75) equal to 1.887458609 and -1.887458609.

Subproblem 1

y + 3.75 = 1.887458609 Simplifying y + 3.75 = 1.887458609 Reorder the terms: 3.75 + y = 1.887458609 Solving 3.75 + y = 1.887458609 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + y = 1.887458609 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + y = 1.887458609 + -3.75 y = 1.887458609 + -3.75 Combine like terms: 1.887458609 + -3.75 = -1.862541391 y = -1.862541391 Simplifying y = -1.862541391

Subproblem 2

y + 3.75 = -1.887458609 Simplifying y + 3.75 = -1.887458609 Reorder the terms: 3.75 + y = -1.887458609 Solving 3.75 + y = -1.887458609 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + y = -1.887458609 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + y = -1.887458609 + -3.75 y = -1.887458609 + -3.75 Combine like terms: -1.887458609 + -3.75 = -5.637458609 y = -5.637458609 Simplifying y = -5.637458609

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-1.862541391, -5.637458609}

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