4x^2+9x-27=0

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


Simplifying
4x2 + 9x + -27 = 0

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

Solving
-27 + 9x + 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'.
-6.75 + 2.25x + x2 = 0

Move the constant term to the right:

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

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

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

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

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = 6.75 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = 6.75 + 1.265625

Combine like terms: 6.75 + 1.265625 = 8.015625
1.265625 + 2.25x + x2 = 8.015625

Factor a perfect square on the left side:
(x + 1.125)(x + 1.125) = 8.015625

Calculate the square root of the right side: 2.831187913

Break this problem into two subproblems by setting 
(x + 1.125) equal to 2.831187913 and -2.831187913.

Subproblem 1

x + 1.125 = 2.831187913 Simplifying x + 1.125 = 2.831187913 Reorder the terms: 1.125 + x = 2.831187913 Solving 1.125 + x = 2.831187913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = 2.831187913 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = 2.831187913 + -1.125 x = 2.831187913 + -1.125 Combine like terms: 2.831187913 + -1.125 = 1.706187913 x = 1.706187913 Simplifying x = 1.706187913

Subproblem 2

x + 1.125 = -2.831187913 Simplifying x + 1.125 = -2.831187913 Reorder the terms: 1.125 + x = -2.831187913 Solving 1.125 + x = -2.831187913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = -2.831187913 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = -2.831187913 + -1.125 x = -2.831187913 + -1.125 Combine like terms: -2.831187913 + -1.125 = -3.956187913 x = -3.956187913 Simplifying x = -3.956187913

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.706187913, -3.956187913}

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