9x^2+5x=6

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Solution for 9x^2+5x=6 equation:


Simplifying
9x2 + 5x = 6

Reorder the terms:
5x + 9x2 = 6

Solving
5x + 9x2 = 6

Solving for variable 'x'.

Reorder the terms:
-6 + 5x + 9x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 5x + 9x2 = 0

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

Divide each side by '9'.
-0.6666666667 + 0.5555555556x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 0.5555555556x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 0.5555555556x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 0.5555555556x + x2 = 0 + 0.6666666667
0.5555555556x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
0.5555555556x + x2 = 0.6666666667

The x term is 0.5555555556x.  Take half its coefficient (0.2777777778).
Square it (0.07716049384) and add it to both sides.

Add '0.07716049384' to each side of the equation.
0.5555555556x + 0.07716049384 + x2 = 0.6666666667 + 0.07716049384

Reorder the terms:
0.07716049384 + 0.5555555556x + x2 = 0.6666666667 + 0.07716049384

Combine like terms: 0.6666666667 + 0.07716049384 = 0.74382716054
0.07716049384 + 0.5555555556x + x2 = 0.74382716054

Factor a perfect square on the left side:
(x + 0.2777777778)(x + 0.2777777778) = 0.74382716054

Calculate the square root of the right side: 0.86245415

Break this problem into two subproblems by setting 
(x + 0.2777777778) equal to 0.86245415 and -0.86245415.

Subproblem 1

x + 0.2777777778 = 0.86245415 Simplifying x + 0.2777777778 = 0.86245415 Reorder the terms: 0.2777777778 + x = 0.86245415 Solving 0.2777777778 + x = 0.86245415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + x = 0.86245415 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + x = 0.86245415 + -0.2777777778 x = 0.86245415 + -0.2777777778 Combine like terms: 0.86245415 + -0.2777777778 = 0.5846763722 x = 0.5846763722 Simplifying x = 0.5846763722

Subproblem 2

x + 0.2777777778 = -0.86245415 Simplifying x + 0.2777777778 = -0.86245415 Reorder the terms: 0.2777777778 + x = -0.86245415 Solving 0.2777777778 + x = -0.86245415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + x = -0.86245415 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + x = -0.86245415 + -0.2777777778 x = -0.86245415 + -0.2777777778 Combine like terms: -0.86245415 + -0.2777777778 = -1.1402319278 x = -1.1402319278 Simplifying x = -1.1402319278

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.5846763722, -1.1402319278}

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