21x^2+20x+3=0

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Solution for 21x^2+20x+3=0 equation:


Simplifying
21x2 + 20x + 3 = 0

Reorder the terms:
3 + 20x + 21x2 = 0

Solving
3 + 20x + 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'.
0.1428571429 + 0.9523809524x + x2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + 0.9523809524x + -0.1428571429 + x2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + 0.9523809524x + x2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + 0.9523809524x + x2 = 0 + -0.1428571429
0.9523809524x + x2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
0.9523809524x + x2 = -0.1428571429

The x term is 0.9523809524x.  Take half its coefficient (0.4761904762).
Square it (0.2267573696) and add it to both sides.

Add '0.2267573696' to each side of the equation.
0.9523809524x + 0.2267573696 + x2 = -0.1428571429 + 0.2267573696

Reorder the terms:
0.2267573696 + 0.9523809524x + x2 = -0.1428571429 + 0.2267573696

Combine like terms: -0.1428571429 + 0.2267573696 = 0.0839002267
0.2267573696 + 0.9523809524x + x2 = 0.0839002267

Factor a perfect square on the left side:
(x + 0.4761904762)(x + 0.4761904762) = 0.0839002267

Calculate the square root of the right side: 0.289655358

Break this problem into two subproblems by setting 
(x + 0.4761904762) equal to 0.289655358 and -0.289655358.

Subproblem 1

x + 0.4761904762 = 0.289655358 Simplifying x + 0.4761904762 = 0.289655358 Reorder the terms: 0.4761904762 + x = 0.289655358 Solving 0.4761904762 + x = 0.289655358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4761904762' to each side of the equation. 0.4761904762 + -0.4761904762 + x = 0.289655358 + -0.4761904762 Combine like terms: 0.4761904762 + -0.4761904762 = 0.0000000000 0.0000000000 + x = 0.289655358 + -0.4761904762 x = 0.289655358 + -0.4761904762 Combine like terms: 0.289655358 + -0.4761904762 = -0.1865351182 x = -0.1865351182 Simplifying x = -0.1865351182

Subproblem 2

x + 0.4761904762 = -0.289655358 Simplifying x + 0.4761904762 = -0.289655358 Reorder the terms: 0.4761904762 + x = -0.289655358 Solving 0.4761904762 + x = -0.289655358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4761904762' to each side of the equation. 0.4761904762 + -0.4761904762 + x = -0.289655358 + -0.4761904762 Combine like terms: 0.4761904762 + -0.4761904762 = 0.0000000000 0.0000000000 + x = -0.289655358 + -0.4761904762 x = -0.289655358 + -0.4761904762 Combine like terms: -0.289655358 + -0.4761904762 = -0.7658458342 x = -0.7658458342 Simplifying x = -0.7658458342

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.1865351182, -0.7658458342}

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