21z^2+3z-20=0

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


Simplifying
21z2 + 3z + -20 = 0

Reorder the terms:
-20 + 3z + 21z2 = 0

Solving
-20 + 3z + 21z2 = 0

Solving for variable 'z'.

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

Divide each side by '21'.
-0.9523809524 + 0.1428571429z + z2 = 0

Move the constant term to the right:

Add '0.9523809524' to each side of the equation.
-0.9523809524 + 0.1428571429z + 0.9523809524 + z2 = 0 + 0.9523809524

Reorder the terms:
-0.9523809524 + 0.9523809524 + 0.1428571429z + z2 = 0 + 0.9523809524

Combine like terms: -0.9523809524 + 0.9523809524 = 0.0000000000
0.0000000000 + 0.1428571429z + z2 = 0 + 0.9523809524
0.1428571429z + z2 = 0 + 0.9523809524

Combine like terms: 0 + 0.9523809524 = 0.9523809524
0.1428571429z + z2 = 0.9523809524

The z term is 0.1428571429z.  Take half its coefficient (0.07142857145).
Square it (0.005102040819) and add it to both sides.

Add '0.005102040819' to each side of the equation.
0.1428571429z + 0.005102040819 + z2 = 0.9523809524 + 0.005102040819

Reorder the terms:
0.005102040819 + 0.1428571429z + z2 = 0.9523809524 + 0.005102040819

Combine like terms: 0.9523809524 + 0.005102040819 = 0.957482993219
0.005102040819 + 0.1428571429z + z2 = 0.957482993219

Factor a perfect square on the left side:
(z + 0.07142857145)(z + 0.07142857145) = 0.957482993219

Calculate the square root of the right side: 0.978510599

Break this problem into two subproblems by setting 
(z + 0.07142857145) equal to 0.978510599 and -0.978510599.

Subproblem 1

z + 0.07142857145 = 0.978510599 Simplifying z + 0.07142857145 = 0.978510599 Reorder the terms: 0.07142857145 + z = 0.978510599 Solving 0.07142857145 + z = 0.978510599 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + z = 0.978510599 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + z = 0.978510599 + -0.07142857145 z = 0.978510599 + -0.07142857145 Combine like terms: 0.978510599 + -0.07142857145 = 0.90708202755 z = 0.90708202755 Simplifying z = 0.90708202755

Subproblem 2

z + 0.07142857145 = -0.978510599 Simplifying z + 0.07142857145 = -0.978510599 Reorder the terms: 0.07142857145 + z = -0.978510599 Solving 0.07142857145 + z = -0.978510599 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + z = -0.978510599 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + z = -0.978510599 + -0.07142857145 z = -0.978510599 + -0.07142857145 Combine like terms: -0.978510599 + -0.07142857145 = -1.04993917045 z = -1.04993917045 Simplifying z = -1.04993917045

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.90708202755, -1.04993917045}

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