9x^2+14x-9=0

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


Simplifying
9x2 + 14x + -9 = 0

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

Solving
-9 + 14x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-1 + 1.555555556x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.555555556x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.555555556x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.555555556x + x2 = 0 + 1
1.555555556x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.555555556x + x2 = 1

The x term is 1.555555556x.  Take half its coefficient (0.777777778).
Square it (0.6049382720) and add it to both sides.

Add '0.6049382720' to each side of the equation.
1.555555556x + 0.6049382720 + x2 = 1 + 0.6049382720

Reorder the terms:
0.6049382720 + 1.555555556x + x2 = 1 + 0.6049382720

Combine like terms: 1 + 0.6049382720 = 1.604938272
0.6049382720 + 1.555555556x + x2 = 1.604938272

Factor a perfect square on the left side:
(x + 0.777777778)(x + 0.777777778) = 1.604938272

Calculate the square root of the right side: 1.266861584

Break this problem into two subproblems by setting 
(x + 0.777777778) equal to 1.266861584 and -1.266861584.

Subproblem 1

x + 0.777777778 = 1.266861584 Simplifying x + 0.777777778 = 1.266861584 Reorder the terms: 0.777777778 + x = 1.266861584 Solving 0.777777778 + x = 1.266861584 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + x = 1.266861584 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + x = 1.266861584 + -0.777777778 x = 1.266861584 + -0.777777778 Combine like terms: 1.266861584 + -0.777777778 = 0.489083806 x = 0.489083806 Simplifying x = 0.489083806

Subproblem 2

x + 0.777777778 = -1.266861584 Simplifying x + 0.777777778 = -1.266861584 Reorder the terms: 0.777777778 + x = -1.266861584 Solving 0.777777778 + x = -1.266861584 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + x = -1.266861584 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + x = -1.266861584 + -0.777777778 x = -1.266861584 + -0.777777778 Combine like terms: -1.266861584 + -0.777777778 = -2.044639362 x = -2.044639362 Simplifying x = -2.044639362

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.489083806, -2.044639362}

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