3*x^2+8*x-9=0

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Solution for 3*x^2+8*x-9=0 equation:


Simplifying
3x2 + 8x + -9 = 0

Reorder the terms:
-9 + 8x + 3x2 = 0

Solving
-9 + 8x + 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 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 2.666666667x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 2.666666667x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 2.666666667x + x2 = 0 + 3
2.666666667x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
2.666666667x + x2 = 3

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = 3 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 3 + 1.777777780

Combine like terms: 3 + 1.777777780 = 4.77777778
1.777777780 + 2.666666667x + x2 = 4.77777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 4.77777778

Calculate the square root of the right side: 2.185812842

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 2.185812842 and -2.185812842.

Subproblem 1

x + 1.333333334 = 2.185812842 Simplifying x + 1.333333334 = 2.185812842 Reorder the terms: 1.333333334 + x = 2.185812842 Solving 1.333333334 + x = 2.185812842 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 2.185812842 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 2.185812842 + -1.333333334 x = 2.185812842 + -1.333333334 Combine like terms: 2.185812842 + -1.333333334 = 0.852479508 x = 0.852479508 Simplifying x = 0.852479508

Subproblem 2

x + 1.333333334 = -2.185812842 Simplifying x + 1.333333334 = -2.185812842 Reorder the terms: 1.333333334 + x = -2.185812842 Solving 1.333333334 + x = -2.185812842 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -2.185812842 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -2.185812842 + -1.333333334 x = -2.185812842 + -1.333333334 Combine like terms: -2.185812842 + -1.333333334 = -3.519146176 x = -3.519146176 Simplifying x = -3.519146176

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.852479508, -3.519146176}

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