3x^2+4x=78

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Solution for 3x^2+4x=78 equation:


Simplifying
3x2 + 4x = 78

Reorder the terms:
4x + 3x2 = 78

Solving
4x + 3x2 = 78

Solving for variable 'x'.

Reorder the terms:
-78 + 4x + 3x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 4x + 3x2 = 0

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

Divide each side by '3'.
-26 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '26' to each side of the equation.
-26 + 1.333333333x + 26 + x2 = 0 + 26

Reorder the terms:
-26 + 26 + 1.333333333x + x2 = 0 + 26

Combine like terms: -26 + 26 = 0
0 + 1.333333333x + x2 = 0 + 26
1.333333333x + x2 = 0 + 26

Combine like terms: 0 + 26 = 26
1.333333333x + x2 = 26

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 26 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 26 + 0.4444444442

Combine like terms: 26 + 0.4444444442 = 26.4444444442
0.4444444442 + 1.333333333x + x2 = 26.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 26.4444444442

Calculate the square root of the right side: 5.142416207

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 5.142416207 and -5.142416207.

Subproblem 1

x + 0.6666666665 = 5.142416207 Simplifying x + 0.6666666665 = 5.142416207 Reorder the terms: 0.6666666665 + x = 5.142416207 Solving 0.6666666665 + x = 5.142416207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 5.142416207 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 5.142416207 + -0.6666666665 x = 5.142416207 + -0.6666666665 Combine like terms: 5.142416207 + -0.6666666665 = 4.4757495405 x = 4.4757495405 Simplifying x = 4.4757495405

Subproblem 2

x + 0.6666666665 = -5.142416207 Simplifying x + 0.6666666665 = -5.142416207 Reorder the terms: 0.6666666665 + x = -5.142416207 Solving 0.6666666665 + x = -5.142416207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -5.142416207 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -5.142416207 + -0.6666666665 x = -5.142416207 + -0.6666666665 Combine like terms: -5.142416207 + -0.6666666665 = -5.8090828735 x = -5.8090828735 Simplifying x = -5.8090828735

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.4757495405, -5.8090828735}

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