4x^2+27x+9=0

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Solution for 4x^2+27x+9=0 equation:


Simplifying
4x2 + 27x + 9 = 0

Reorder the terms:
9 + 27x + 4x2 = 0

Solving
9 + 27x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
2.25 + 6.75x + x2 = 0

Move the constant term to the right:

Add '-2.25' to each side of the equation.
2.25 + 6.75x + -2.25 + x2 = 0 + -2.25

Reorder the terms:
2.25 + -2.25 + 6.75x + x2 = 0 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + 6.75x + x2 = 0 + -2.25
6.75x + x2 = 0 + -2.25

Combine like terms: 0 + -2.25 = -2.25
6.75x + x2 = -2.25

The x term is 6.75x.  Take half its coefficient (3.375).
Square it (11.390625) and add it to both sides.

Add '11.390625' to each side of the equation.
6.75x + 11.390625 + x2 = -2.25 + 11.390625

Reorder the terms:
11.390625 + 6.75x + x2 = -2.25 + 11.390625

Combine like terms: -2.25 + 11.390625 = 9.140625
11.390625 + 6.75x + x2 = 9.140625

Factor a perfect square on the left side:
(x + 3.375)(x + 3.375) = 9.140625

Calculate the square root of the right side: 3.023346656

Break this problem into two subproblems by setting 
(x + 3.375) equal to 3.023346656 and -3.023346656.

Subproblem 1

x + 3.375 = 3.023346656 Simplifying x + 3.375 = 3.023346656 Reorder the terms: 3.375 + x = 3.023346656 Solving 3.375 + x = 3.023346656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.375' to each side of the equation. 3.375 + -3.375 + x = 3.023346656 + -3.375 Combine like terms: 3.375 + -3.375 = 0.000 0.000 + x = 3.023346656 + -3.375 x = 3.023346656 + -3.375 Combine like terms: 3.023346656 + -3.375 = -0.351653344 x = -0.351653344 Simplifying x = -0.351653344

Subproblem 2

x + 3.375 = -3.023346656 Simplifying x + 3.375 = -3.023346656 Reorder the terms: 3.375 + x = -3.023346656 Solving 3.375 + x = -3.023346656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.375' to each side of the equation. 3.375 + -3.375 + x = -3.023346656 + -3.375 Combine like terms: 3.375 + -3.375 = 0.000 0.000 + x = -3.023346656 + -3.375 x = -3.023346656 + -3.375 Combine like terms: -3.023346656 + -3.375 = -6.398346656 x = -6.398346656 Simplifying x = -6.398346656

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.351653344, -6.398346656}

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