(x)(3x+2)=420

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Solution for (x)(3x+2)=420 equation:


Simplifying
(x)(3x + 2) = 420

Reorder the terms:
x(2 + 3x) = 420
(2 * x + 3x * x) = 420
(2x + 3x2) = 420

Solving
2x + 3x2 = 420

Solving for variable 'x'.

Reorder the terms:
-420 + 2x + 3x2 = 420 + -420

Combine like terms: 420 + -420 = 0
-420 + 2x + 3x2 = 0

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

Divide each side by '3'.
-140 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '140' to each side of the equation.
-140 + 0.6666666667x + 140 + x2 = 0 + 140

Reorder the terms:
-140 + 140 + 0.6666666667x + x2 = 0 + 140

Combine like terms: -140 + 140 = 0
0 + 0.6666666667x + x2 = 0 + 140
0.6666666667x + x2 = 0 + 140

Combine like terms: 0 + 140 = 140
0.6666666667x + x2 = 140

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 = 140 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 140 + 0.1111111112

Combine like terms: 140 + 0.1111111112 = 140.1111111112
0.1111111112 + 0.6666666667x + x2 = 140.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 140.1111111112

Calculate the square root of the right side: 11.836853936

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 11.836853936 and -11.836853936.

Subproblem 1

x + 0.3333333334 = 11.836853936 Simplifying x + 0.3333333334 = 11.836853936 Reorder the terms: 0.3333333334 + x = 11.836853936 Solving 0.3333333334 + x = 11.836853936 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 = 11.836853936 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 11.836853936 + -0.3333333334 x = 11.836853936 + -0.3333333334 Combine like terms: 11.836853936 + -0.3333333334 = 11.5035206026 x = 11.5035206026 Simplifying x = 11.5035206026

Subproblem 2

x + 0.3333333334 = -11.836853936 Simplifying x + 0.3333333334 = -11.836853936 Reorder the terms: 0.3333333334 + x = -11.836853936 Solving 0.3333333334 + x = -11.836853936 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 = -11.836853936 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -11.836853936 + -0.3333333334 x = -11.836853936 + -0.3333333334 Combine like terms: -11.836853936 + -0.3333333334 = -12.1701872694 x = -12.1701872694 Simplifying x = -12.1701872694

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.5035206026, -12.1701872694}

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