6x^2+32x+32=71

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Solution for 6x^2+32x+32=71 equation:


Simplifying
6x2 + 32x + 32 = 71

Reorder the terms:
32 + 32x + 6x2 = 71

Solving
32 + 32x + 6x2 = 71

Solving for variable 'x'.

Reorder the terms:
32 + -71 + 32x + 6x2 = 71 + -71

Combine like terms: 32 + -71 = -39
-39 + 32x + 6x2 = 71 + -71

Combine like terms: 71 + -71 = 0
-39 + 32x + 6x2 = 0

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

Divide each side by '6'.
-6.5 + 5.333333333x + x2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + 5.333333333x + 6.5 + x2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + 5.333333333x + x2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + 5.333333333x + x2 = 0 + 6.5
5.333333333x + x2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
5.333333333x + x2 = 6.5

The x term is 5.333333333x.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333x + 7.111111113 + x2 = 6.5 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333x + x2 = 6.5 + 7.111111113

Combine like terms: 6.5 + 7.111111113 = 13.611111113
7.111111113 + 5.333333333x + x2 = 13.611111113

Factor a perfect square on the left side:
(x + 2.666666667)(x + 2.666666667) = 13.611111113

Calculate the square root of the right side: 3.689323937

Break this problem into two subproblems by setting 
(x + 2.666666667) equal to 3.689323937 and -3.689323937.

Subproblem 1

x + 2.666666667 = 3.689323937 Simplifying x + 2.666666667 = 3.689323937 Reorder the terms: 2.666666667 + x = 3.689323937 Solving 2.666666667 + x = 3.689323937 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = 3.689323937 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = 3.689323937 + -2.666666667 x = 3.689323937 + -2.666666667 Combine like terms: 3.689323937 + -2.666666667 = 1.02265727 x = 1.02265727 Simplifying x = 1.02265727

Subproblem 2

x + 2.666666667 = -3.689323937 Simplifying x + 2.666666667 = -3.689323937 Reorder the terms: 2.666666667 + x = -3.689323937 Solving 2.666666667 + x = -3.689323937 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = -3.689323937 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = -3.689323937 + -2.666666667 x = -3.689323937 + -2.666666667 Combine like terms: -3.689323937 + -2.666666667 = -6.355990604 x = -6.355990604 Simplifying x = -6.355990604

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.02265727, -6.355990604}

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