21x^2-23x-6=0

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Solution for 21x^2-23x-6=0 equation:


Simplifying
21x2 + -23x + -6 = 0

Reorder the terms:
-6 + -23x + 21x2 = 0

Solving
-6 + -23x + 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'.
-0.2857142857 + -1.095238095x + x2 = 0

Move the constant term to the right:

Add '0.2857142857' to each side of the equation.
-0.2857142857 + -1.095238095x + 0.2857142857 + x2 = 0 + 0.2857142857

Reorder the terms:
-0.2857142857 + 0.2857142857 + -1.095238095x + x2 = 0 + 0.2857142857

Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000
0.0000000000 + -1.095238095x + x2 = 0 + 0.2857142857
-1.095238095x + x2 = 0 + 0.2857142857

Combine like terms: 0 + 0.2857142857 = 0.2857142857
-1.095238095x + x2 = 0.2857142857

The x term is -1.095238095x.  Take half its coefficient (-0.5476190475).
Square it (0.2998866212) and add it to both sides.

Add '0.2998866212' to each side of the equation.
-1.095238095x + 0.2998866212 + x2 = 0.2857142857 + 0.2998866212

Reorder the terms:
0.2998866212 + -1.095238095x + x2 = 0.2857142857 + 0.2998866212

Combine like terms: 0.2857142857 + 0.2998866212 = 0.5856009069
0.2998866212 + -1.095238095x + x2 = 0.5856009069

Factor a perfect square on the left side:
(x + -0.5476190475)(x + -0.5476190475) = 0.5856009069

Calculate the square root of the right side: 0.765245651

Break this problem into two subproblems by setting 
(x + -0.5476190475) equal to 0.765245651 and -0.765245651.

Subproblem 1

x + -0.5476190475 = 0.765245651 Simplifying x + -0.5476190475 = 0.765245651 Reorder the terms: -0.5476190475 + x = 0.765245651 Solving -0.5476190475 + x = 0.765245651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5476190475' to each side of the equation. -0.5476190475 + 0.5476190475 + x = 0.765245651 + 0.5476190475 Combine like terms: -0.5476190475 + 0.5476190475 = 0.0000000000 0.0000000000 + x = 0.765245651 + 0.5476190475 x = 0.765245651 + 0.5476190475 Combine like terms: 0.765245651 + 0.5476190475 = 1.3128646985 x = 1.3128646985 Simplifying x = 1.3128646985

Subproblem 2

x + -0.5476190475 = -0.765245651 Simplifying x + -0.5476190475 = -0.765245651 Reorder the terms: -0.5476190475 + x = -0.765245651 Solving -0.5476190475 + x = -0.765245651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5476190475' to each side of the equation. -0.5476190475 + 0.5476190475 + x = -0.765245651 + 0.5476190475 Combine like terms: -0.5476190475 + 0.5476190475 = 0.0000000000 0.0000000000 + x = -0.765245651 + 0.5476190475 x = -0.765245651 + 0.5476190475 Combine like terms: -0.765245651 + 0.5476190475 = -0.2176266035 x = -0.2176266035 Simplifying x = -0.2176266035

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3128646985, -0.2176266035}

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