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