12x^2+5x=84

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Solution for 12x^2+5x=84 equation:


Simplifying
12x2 + 5x = 84

Reorder the terms:
5x + 12x2 = 84

Solving
5x + 12x2 = 84

Solving for variable 'x'.

Reorder the terms:
-84 + 5x + 12x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 5x + 12x2 = 0

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

Divide each side by '12'.
-7 + 0.4166666667x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 0.4166666667x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + 0.4166666667x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 0.4166666667x + x2 = 0 + 7
0.4166666667x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
0.4166666667x + x2 = 7

The x term is 0.4166666667x.  Take half its coefficient (0.2083333334).
Square it (0.04340277781) and add it to both sides.

Add '0.04340277781' to each side of the equation.
0.4166666667x + 0.04340277781 + x2 = 7 + 0.04340277781

Reorder the terms:
0.04340277781 + 0.4166666667x + x2 = 7 + 0.04340277781

Combine like terms: 7 + 0.04340277781 = 7.04340277781
0.04340277781 + 0.4166666667x + x2 = 7.04340277781

Factor a perfect square on the left side:
(x + 0.2083333334)(x + 0.2083333334) = 7.04340277781

Calculate the square root of the right side: 2.65394099

Break this problem into two subproblems by setting 
(x + 0.2083333334) equal to 2.65394099 and -2.65394099.

Subproblem 1

x + 0.2083333334 = 2.65394099 Simplifying x + 0.2083333334 = 2.65394099 Reorder the terms: 0.2083333334 + x = 2.65394099 Solving 0.2083333334 + x = 2.65394099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2083333334' to each side of the equation. 0.2083333334 + -0.2083333334 + x = 2.65394099 + -0.2083333334 Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000 0.0000000000 + x = 2.65394099 + -0.2083333334 x = 2.65394099 + -0.2083333334 Combine like terms: 2.65394099 + -0.2083333334 = 2.4456076566 x = 2.4456076566 Simplifying x = 2.4456076566

Subproblem 2

x + 0.2083333334 = -2.65394099 Simplifying x + 0.2083333334 = -2.65394099 Reorder the terms: 0.2083333334 + x = -2.65394099 Solving 0.2083333334 + x = -2.65394099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2083333334' to each side of the equation. 0.2083333334 + -0.2083333334 + x = -2.65394099 + -0.2083333334 Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000 0.0000000000 + x = -2.65394099 + -0.2083333334 x = -2.65394099 + -0.2083333334 Combine like terms: -2.65394099 + -0.2083333334 = -2.8622743234 x = -2.8622743234 Simplifying x = -2.8622743234

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.4456076566, -2.8622743234}

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