0=3x^2+5x-9

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Solution for 0=3x^2+5x-9 equation:


Simplifying
0 = 3x2 + 5x + -9

Reorder the terms:
0 = -9 + 5x + 3x2

Solving
0 = -9 + 5x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 9 = 9
9 + -5x + -3x2 = -9 + 5x + 3x2 + 9 + -5x + -3x2

Reorder the terms:
9 + -5x + -3x2 = -9 + 9 + 5x + -5x + 3x2 + -3x2

Combine like terms: -9 + 9 = 0
9 + -5x + -3x2 = 0 + 5x + -5x + 3x2 + -3x2
9 + -5x + -3x2 = 5x + -5x + 3x2 + -3x2

Combine like terms: 5x + -5x = 0
9 + -5x + -3x2 = 0 + 3x2 + -3x2
9 + -5x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
9 + -5x + -3x2 = 0

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

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

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 1.666666667x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 1.666666667x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 1.666666667x + x2 = 0 + 3
1.666666667x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
1.666666667x + x2 = 3

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 = 3 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 3 + 0.6944444447

Combine like terms: 3 + 0.6944444447 = 3.6944444447
0.6944444447 + 1.666666667x + x2 = 3.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 3.6944444447

Calculate the square root of the right side: 1.922093766

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 1.922093766 and -1.922093766.

Subproblem 1

x + 0.8333333335 = 1.922093766 Simplifying x + 0.8333333335 = 1.922093766 Reorder the terms: 0.8333333335 + x = 1.922093766 Solving 0.8333333335 + x = 1.922093766 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 = 1.922093766 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 1.922093766 + -0.8333333335 x = 1.922093766 + -0.8333333335 Combine like terms: 1.922093766 + -0.8333333335 = 1.0887604325 x = 1.0887604325 Simplifying x = 1.0887604325

Subproblem 2

x + 0.8333333335 = -1.922093766 Simplifying x + 0.8333333335 = -1.922093766 Reorder the terms: 0.8333333335 + x = -1.922093766 Solving 0.8333333335 + x = -1.922093766 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 = -1.922093766 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -1.922093766 + -0.8333333335 x = -1.922093766 + -0.8333333335 Combine like terms: -1.922093766 + -0.8333333335 = -2.7554270995 x = -2.7554270995 Simplifying x = -2.7554270995

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0887604325, -2.7554270995}

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