3x^2+2x-78=0

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Solution for 3x^2+2x-78=0 equation:


Simplifying
3x2 + 2x + -78 = 0

Reorder the terms:
-78 + 2x + 3x2 = 0

Solving
-78 + 2x + 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'.
-26 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '26' to each side of the equation.
-26 + 0.6666666667x + 26 + x2 = 0 + 26

Reorder the terms:
-26 + 26 + 0.6666666667x + x2 = 0 + 26

Combine like terms: -26 + 26 = 0
0 + 0.6666666667x + x2 = 0 + 26
0.6666666667x + x2 = 0 + 26

Combine like terms: 0 + 26 = 26
0.6666666667x + x2 = 26

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 26 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 26 + 0.1111111112

Combine like terms: 26 + 0.1111111112 = 26.1111111112
0.1111111112 + 0.6666666667x + x2 = 26.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 26.1111111112

Calculate the square root of the right side: 5.109903239

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 5.109903239 and -5.109903239.

Subproblem 1

x + 0.3333333334 = 5.109903239 Simplifying x + 0.3333333334 = 5.109903239 Reorder the terms: 0.3333333334 + x = 5.109903239 Solving 0.3333333334 + x = 5.109903239 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 5.109903239 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 5.109903239 + -0.3333333334 x = 5.109903239 + -0.3333333334 Combine like terms: 5.109903239 + -0.3333333334 = 4.7765699056 x = 4.7765699056 Simplifying x = 4.7765699056

Subproblem 2

x + 0.3333333334 = -5.109903239 Simplifying x + 0.3333333334 = -5.109903239 Reorder the terms: 0.3333333334 + x = -5.109903239 Solving 0.3333333334 + x = -5.109903239 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -5.109903239 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -5.109903239 + -0.3333333334 x = -5.109903239 + -0.3333333334 Combine like terms: -5.109903239 + -0.3333333334 = -5.4432365724 x = -5.4432365724 Simplifying x = -5.4432365724

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.7765699056, -5.4432365724}

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