3x^2+14x-9=0

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Solution for 3x^2+14x-9=0 equation:


Simplifying
3x2 + 14x + -9 = 0

Reorder the terms:
-9 + 14x + 3x2 = 0

Solving
-9 + 14x + 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'.
-3 + 4.666666667x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 4.666666667x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 4.666666667x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 4.666666667x + x2 = 0 + 3
4.666666667x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
4.666666667x + x2 = 3

The x term is 4.666666667x.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667x + 5.444444448 + x2 = 3 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667x + x2 = 3 + 5.444444448

Combine like terms: 3 + 5.444444448 = 8.444444448
5.444444448 + 4.666666667x + x2 = 8.444444448

Factor a perfect square on the left side:
(x + 2.333333334)(x + 2.333333334) = 8.444444448

Calculate the square root of the right side: 2.90593263

Break this problem into two subproblems by setting 
(x + 2.333333334) equal to 2.90593263 and -2.90593263.

Subproblem 1

x + 2.333333334 = 2.90593263 Simplifying x + 2.333333334 = 2.90593263 Reorder the terms: 2.333333334 + x = 2.90593263 Solving 2.333333334 + x = 2.90593263 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = 2.90593263 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = 2.90593263 + -2.333333334 x = 2.90593263 + -2.333333334 Combine like terms: 2.90593263 + -2.333333334 = 0.572599296 x = 0.572599296 Simplifying x = 0.572599296

Subproblem 2

x + 2.333333334 = -2.90593263 Simplifying x + 2.333333334 = -2.90593263 Reorder the terms: 2.333333334 + x = -2.90593263 Solving 2.333333334 + x = -2.90593263 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = -2.90593263 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = -2.90593263 + -2.333333334 x = -2.90593263 + -2.333333334 Combine like terms: -2.90593263 + -2.333333334 = -5.239265964 x = -5.239265964 Simplifying x = -5.239265964

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.572599296, -5.239265964}

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