4y^2+24y+9=0

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Solution for 4y^2+24y+9=0 equation:


Simplifying
4y2 + 24y + 9 = 0

Reorder the terms:
9 + 24y + 4y2 = 0

Solving
9 + 24y + 4y2 = 0

Solving for variable 'y'.

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

Divide each side by '4'.
2.25 + 6y + y2 = 0

Move the constant term to the right:

Add '-2.25' to each side of the equation.
2.25 + 6y + -2.25 + y2 = 0 + -2.25

Reorder the terms:
2.25 + -2.25 + 6y + y2 = 0 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + 6y + y2 = 0 + -2.25
6y + y2 = 0 + -2.25

Combine like terms: 0 + -2.25 = -2.25
6y + y2 = -2.25

The y term is 6y.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6y + 9 + y2 = -2.25 + 9

Reorder the terms:
9 + 6y + y2 = -2.25 + 9

Combine like terms: -2.25 + 9 = 6.75
9 + 6y + y2 = 6.75

Factor a perfect square on the left side:
(y + 3)(y + 3) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(y + 3) equal to 2.598076211 and -2.598076211.

Subproblem 1

y + 3 = 2.598076211 Simplifying y + 3 = 2.598076211 Reorder the terms: 3 + y = 2.598076211 Solving 3 + y = 2.598076211 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + y = 2.598076211 + -3 Combine like terms: 3 + -3 = 0 0 + y = 2.598076211 + -3 y = 2.598076211 + -3 Combine like terms: 2.598076211 + -3 = -0.401923789 y = -0.401923789 Simplifying y = -0.401923789

Subproblem 2

y + 3 = -2.598076211 Simplifying y + 3 = -2.598076211 Reorder the terms: 3 + y = -2.598076211 Solving 3 + y = -2.598076211 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + y = -2.598076211 + -3 Combine like terms: 3 + -3 = 0 0 + y = -2.598076211 + -3 y = -2.598076211 + -3 Combine like terms: -2.598076211 + -3 = -5.598076211 y = -5.598076211 Simplifying y = -5.598076211

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.401923789, -5.598076211}

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