21x^2+11x-86=0

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Solution for 21x^2+11x-86=0 equation:


Simplifying
21x2 + 11x + -86 = 0

Reorder the terms:
-86 + 11x + 21x2 = 0

Solving
-86 + 11x + 21x2 = 0

Solving for variable 'x'.

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

Divide each side by '21'.
-4.095238095 + 0.5238095238x + x2 = 0

Move the constant term to the right:

Add '4.095238095' to each side of the equation.
-4.095238095 + 0.5238095238x + 4.095238095 + x2 = 0 + 4.095238095

Reorder the terms:
-4.095238095 + 4.095238095 + 0.5238095238x + x2 = 0 + 4.095238095

Combine like terms: -4.095238095 + 4.095238095 = 0.000000000
0.000000000 + 0.5238095238x + x2 = 0 + 4.095238095
0.5238095238x + x2 = 0 + 4.095238095

Combine like terms: 0 + 4.095238095 = 4.095238095
0.5238095238x + x2 = 4.095238095

The x term is 0.5238095238x.  Take half its coefficient (0.2619047619).
Square it (0.06859410431) and add it to both sides.

Add '0.06859410431' to each side of the equation.
0.5238095238x + 0.06859410431 + x2 = 4.095238095 + 0.06859410431

Reorder the terms:
0.06859410431 + 0.5238095238x + x2 = 4.095238095 + 0.06859410431

Combine like terms: 4.095238095 + 0.06859410431 = 4.16383219931
0.06859410431 + 0.5238095238x + x2 = 4.16383219931

Factor a perfect square on the left side:
(x + 0.2619047619)(x + 0.2619047619) = 4.16383219931

Calculate the square root of the right side: 2.040547034

Break this problem into two subproblems by setting 
(x + 0.2619047619) equal to 2.040547034 and -2.040547034.

Subproblem 1

x + 0.2619047619 = 2.040547034 Simplifying x + 0.2619047619 = 2.040547034 Reorder the terms: 0.2619047619 + x = 2.040547034 Solving 0.2619047619 + x = 2.040547034 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2619047619' to each side of the equation. 0.2619047619 + -0.2619047619 + x = 2.040547034 + -0.2619047619 Combine like terms: 0.2619047619 + -0.2619047619 = 0.0000000000 0.0000000000 + x = 2.040547034 + -0.2619047619 x = 2.040547034 + -0.2619047619 Combine like terms: 2.040547034 + -0.2619047619 = 1.7786422721 x = 1.7786422721 Simplifying x = 1.7786422721

Subproblem 2

x + 0.2619047619 = -2.040547034 Simplifying x + 0.2619047619 = -2.040547034 Reorder the terms: 0.2619047619 + x = -2.040547034 Solving 0.2619047619 + x = -2.040547034 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2619047619' to each side of the equation. 0.2619047619 + -0.2619047619 + x = -2.040547034 + -0.2619047619 Combine like terms: 0.2619047619 + -0.2619047619 = 0.0000000000 0.0000000000 + x = -2.040547034 + -0.2619047619 x = -2.040547034 + -0.2619047619 Combine like terms: -2.040547034 + -0.2619047619 = -2.3024517959 x = -2.3024517959 Simplifying x = -2.3024517959

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.7786422721, -2.3024517959}

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