3x^2+23x-9=0

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Solution for 3x^2+23x-9=0 equation:


Simplifying
3x2 + 23x + -9 = 0

Reorder the terms:
-9 + 23x + 3x2 = 0

Solving
-9 + 23x + 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 + 7.666666667x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 7.666666667x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 7.666666667x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 7.666666667x + x2 = 0 + 3
7.666666667x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
7.666666667x + x2 = 3

The x term is 7.666666667x.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667x + 14.69444445 + x2 = 3 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667x + x2 = 3 + 14.69444445

Combine like terms: 3 + 14.69444445 = 17.69444445
14.69444445 + 7.666666667x + x2 = 17.69444445

Factor a perfect square on the left side:
(x + 3.833333334)(x + 3.833333334) = 17.69444445

Calculate the square root of the right side: 4.206476489

Break this problem into two subproblems by setting 
(x + 3.833333334) equal to 4.206476489 and -4.206476489.

Subproblem 1

x + 3.833333334 = 4.206476489 Simplifying x + 3.833333334 = 4.206476489 Reorder the terms: 3.833333334 + x = 4.206476489 Solving 3.833333334 + x = 4.206476489 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + x = 4.206476489 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + x = 4.206476489 + -3.833333334 x = 4.206476489 + -3.833333334 Combine like terms: 4.206476489 + -3.833333334 = 0.373143155 x = 0.373143155 Simplifying x = 0.373143155

Subproblem 2

x + 3.833333334 = -4.206476489 Simplifying x + 3.833333334 = -4.206476489 Reorder the terms: 3.833333334 + x = -4.206476489 Solving 3.833333334 + x = -4.206476489 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + x = -4.206476489 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + x = -4.206476489 + -3.833333334 x = -4.206476489 + -3.833333334 Combine like terms: -4.206476489 + -3.833333334 = -8.039809823 x = -8.039809823 Simplifying x = -8.039809823

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.373143155, -8.039809823}

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