3x^2+16x-84=0

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Solution for 3x^2+16x-84=0 equation:


Simplifying
3x2 + 16x + -84 = 0

Reorder the terms:
-84 + 16x + 3x2 = 0

Solving
-84 + 16x + 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'.
-28 + 5.333333333x + x2 = 0

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + 5.333333333x + 28 + x2 = 0 + 28

Reorder the terms:
-28 + 28 + 5.333333333x + x2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + 5.333333333x + x2 = 0 + 28
5.333333333x + x2 = 0 + 28

Combine like terms: 0 + 28 = 28
5.333333333x + x2 = 28

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 = 28 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333x + x2 = 28 + 7.111111113

Combine like terms: 28 + 7.111111113 = 35.111111113
7.111111113 + 5.333333333x + x2 = 35.111111113

Factor a perfect square on the left side:
(x + 2.666666667)(x + 2.666666667) = 35.111111113

Calculate the square root of the right side: 5.925462945

Break this problem into two subproblems by setting 
(x + 2.666666667) equal to 5.925462945 and -5.925462945.

Subproblem 1

x + 2.666666667 = 5.925462945 Simplifying x + 2.666666667 = 5.925462945 Reorder the terms: 2.666666667 + x = 5.925462945 Solving 2.666666667 + x = 5.925462945 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 = 5.925462945 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = 5.925462945 + -2.666666667 x = 5.925462945 + -2.666666667 Combine like terms: 5.925462945 + -2.666666667 = 3.258796278 x = 3.258796278 Simplifying x = 3.258796278

Subproblem 2

x + 2.666666667 = -5.925462945 Simplifying x + 2.666666667 = -5.925462945 Reorder the terms: 2.666666667 + x = -5.925462945 Solving 2.666666667 + x = -5.925462945 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 = -5.925462945 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = -5.925462945 + -2.666666667 x = -5.925462945 + -2.666666667 Combine like terms: -5.925462945 + -2.666666667 = -8.592129612 x = -8.592129612 Simplifying x = -8.592129612

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.258796278, -8.592129612}

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