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Simplifying 7z2 + 8z + -5 = 0 Reorder the terms: -5 + 8z + 7z2 = 0 Solving -5 + 8z + 7z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.7142857143 + 1.142857143z + z2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 1.142857143z + 0.7142857143 + z2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 1.142857143z + z2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 1.142857143z + z2 = 0 + 0.7142857143 1.142857143z + z2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 1.142857143z + z2 = 0.7142857143 The z term is 1.142857143z. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143z + 0.3265306123 + z2 = 0.7142857143 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143z + z2 = 0.7142857143 + 0.3265306123 Combine like terms: 0.7142857143 + 0.3265306123 = 1.0408163266 0.3265306123 + 1.142857143z + z2 = 1.0408163266 Factor a perfect square on the left side: (z + 0.5714285715)(z + 0.5714285715) = 1.0408163266 Calculate the square root of the right side: 1.020204061 Break this problem into two subproblems by setting (z + 0.5714285715) equal to 1.020204061 and -1.020204061.Subproblem 1
z + 0.5714285715 = 1.020204061 Simplifying z + 0.5714285715 = 1.020204061 Reorder the terms: 0.5714285715 + z = 1.020204061 Solving 0.5714285715 + z = 1.020204061 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + z = 1.020204061 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + z = 1.020204061 + -0.5714285715 z = 1.020204061 + -0.5714285715 Combine like terms: 1.020204061 + -0.5714285715 = 0.4487754895 z = 0.4487754895 Simplifying z = 0.4487754895Subproblem 2
z + 0.5714285715 = -1.020204061 Simplifying z + 0.5714285715 = -1.020204061 Reorder the terms: 0.5714285715 + z = -1.020204061 Solving 0.5714285715 + z = -1.020204061 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + z = -1.020204061 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + z = -1.020204061 + -0.5714285715 z = -1.020204061 + -0.5714285715 Combine like terms: -1.020204061 + -0.5714285715 = -1.5916326325 z = -1.5916326325 Simplifying z = -1.5916326325Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.4487754895, -1.5916326325}
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