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Simplifying 13z2 + -16z + -4 = 0 Reorder the terms: -4 + -16z + 13z2 = 0 Solving -4 + -16z + 13z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -0.3076923077 + -1.230769231z + z2 = 0 Move the constant term to the right: Add '0.3076923077' to each side of the equation. -0.3076923077 + -1.230769231z + 0.3076923077 + z2 = 0 + 0.3076923077 Reorder the terms: -0.3076923077 + 0.3076923077 + -1.230769231z + z2 = 0 + 0.3076923077 Combine like terms: -0.3076923077 + 0.3076923077 = 0.0000000000 0.0000000000 + -1.230769231z + z2 = 0 + 0.3076923077 -1.230769231z + z2 = 0 + 0.3076923077 Combine like terms: 0 + 0.3076923077 = 0.3076923077 -1.230769231z + z2 = 0.3076923077 The z term is -1.230769231z. Take half its coefficient (-0.6153846155). Square it (0.3786982250) and add it to both sides. Add '0.3786982250' to each side of the equation. -1.230769231z + 0.3786982250 + z2 = 0.3076923077 + 0.3786982250 Reorder the terms: 0.3786982250 + -1.230769231z + z2 = 0.3076923077 + 0.3786982250 Combine like terms: 0.3076923077 + 0.3786982250 = 0.6863905327 0.3786982250 + -1.230769231z + z2 = 0.6863905327 Factor a perfect square on the left side: (z + -0.6153846155)(z + -0.6153846155) = 0.6863905327 Calculate the square root of the right side: 0.828486893 Break this problem into two subproblems by setting (z + -0.6153846155) equal to 0.828486893 and -0.828486893.Subproblem 1
z + -0.6153846155 = 0.828486893 Simplifying z + -0.6153846155 = 0.828486893 Reorder the terms: -0.6153846155 + z = 0.828486893 Solving -0.6153846155 + z = 0.828486893 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6153846155' to each side of the equation. -0.6153846155 + 0.6153846155 + z = 0.828486893 + 0.6153846155 Combine like terms: -0.6153846155 + 0.6153846155 = 0.0000000000 0.0000000000 + z = 0.828486893 + 0.6153846155 z = 0.828486893 + 0.6153846155 Combine like terms: 0.828486893 + 0.6153846155 = 1.4438715085 z = 1.4438715085 Simplifying z = 1.4438715085Subproblem 2
z + -0.6153846155 = -0.828486893 Simplifying z + -0.6153846155 = -0.828486893 Reorder the terms: -0.6153846155 + z = -0.828486893 Solving -0.6153846155 + z = -0.828486893 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6153846155' to each side of the equation. -0.6153846155 + 0.6153846155 + z = -0.828486893 + 0.6153846155 Combine like terms: -0.6153846155 + 0.6153846155 = 0.0000000000 0.0000000000 + z = -0.828486893 + 0.6153846155 z = -0.828486893 + 0.6153846155 Combine like terms: -0.828486893 + 0.6153846155 = -0.2131022775 z = -0.2131022775 Simplifying z = -0.2131022775Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.4438715085, -0.2131022775}
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