3x^2+9x-61=0

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Solution for 3x^2+9x-61=0 equation:


Simplifying
3x2 + 9x + -61 = 0

Reorder the terms:
-61 + 9x + 3x2 = 0

Solving
-61 + 9x + 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'.
-20.33333333 + 3x + x2 = 0

Move the constant term to the right:

Add '20.33333333' to each side of the equation.
-20.33333333 + 3x + 20.33333333 + x2 = 0 + 20.33333333

Reorder the terms:
-20.33333333 + 20.33333333 + 3x + x2 = 0 + 20.33333333

Combine like terms: -20.33333333 + 20.33333333 = 0.00000000
0.00000000 + 3x + x2 = 0 + 20.33333333
3x + x2 = 0 + 20.33333333

Combine like terms: 0 + 20.33333333 = 20.33333333
3x + x2 = 20.33333333

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 20.33333333 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 20.33333333 + 2.25

Combine like terms: 20.33333333 + 2.25 = 22.58333333
2.25 + 3x + x2 = 22.58333333

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 22.58333333

Calculate the square root of the right side: 4.752192476

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.752192476 and -4.752192476.

Subproblem 1

x + 1.5 = 4.752192476 Simplifying x + 1.5 = 4.752192476 Reorder the terms: 1.5 + x = 4.752192476 Solving 1.5 + x = 4.752192476 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 4.752192476 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.752192476 + -1.5 x = 4.752192476 + -1.5 Combine like terms: 4.752192476 + -1.5 = 3.252192476 x = 3.252192476 Simplifying x = 3.252192476

Subproblem 2

x + 1.5 = -4.752192476 Simplifying x + 1.5 = -4.752192476 Reorder the terms: 1.5 + x = -4.752192476 Solving 1.5 + x = -4.752192476 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -4.752192476 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.752192476 + -1.5 x = -4.752192476 + -1.5 Combine like terms: -4.752192476 + -1.5 = -6.252192476 x = -6.252192476 Simplifying x = -6.252192476

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.252192476, -6.252192476}

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