3x^2+5x-27=0

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Solution for 3x^2+5x-27=0 equation:


Simplifying
3x2 + 5x + -27 = 0

Reorder the terms:
-27 + 5x + 3x2 = 0

Solving
-27 + 5x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-9 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 1.666666667x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + 1.666666667x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 1.666666667x + x2 = 0 + 9
1.666666667x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
1.666666667x + x2 = 9

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 9 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 9 + 0.6944444447

Combine like terms: 9 + 0.6944444447 = 9.6944444447
0.6944444447 + 1.666666667x + x2 = 9.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 9.6944444447

Calculate the square root of the right side: 3.113590282

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 3.113590282 and -3.113590282.

Subproblem 1

x + 0.8333333335 = 3.113590282 Simplifying x + 0.8333333335 = 3.113590282 Reorder the terms: 0.8333333335 + x = 3.113590282 Solving 0.8333333335 + x = 3.113590282 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = 3.113590282 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 3.113590282 + -0.8333333335 x = 3.113590282 + -0.8333333335 Combine like terms: 3.113590282 + -0.8333333335 = 2.2802569485 x = 2.2802569485 Simplifying x = 2.2802569485

Subproblem 2

x + 0.8333333335 = -3.113590282 Simplifying x + 0.8333333335 = -3.113590282 Reorder the terms: 0.8333333335 + x = -3.113590282 Solving 0.8333333335 + x = -3.113590282 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = -3.113590282 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -3.113590282 + -0.8333333335 x = -3.113590282 + -0.8333333335 Combine like terms: -3.113590282 + -0.8333333335 = -3.9469236155 x = -3.9469236155 Simplifying x = -3.9469236155

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.2802569485, -3.9469236155}

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