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Simplifying 4z2 + -16z + 13 = 0 Reorder the terms: 13 + -16z + 4z2 = 0 Solving 13 + -16z + 4z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 3.25 + -4z + z2 = 0 Move the constant term to the right: Add '-3.25' to each side of the equation. 3.25 + -4z + -3.25 + z2 = 0 + -3.25 Reorder the terms: 3.25 + -3.25 + -4z + z2 = 0 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + -4z + z2 = 0 + -3.25 -4z + z2 = 0 + -3.25 Combine like terms: 0 + -3.25 = -3.25 -4z + z2 = -3.25 The z term is -4z. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4z + 4 + z2 = -3.25 + 4 Reorder the terms: 4 + -4z + z2 = -3.25 + 4 Combine like terms: -3.25 + 4 = 0.75 4 + -4z + z2 = 0.75 Factor a perfect square on the left side: (z + -2)(z + -2) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (z + -2) equal to 0.866025404 and -0.866025404.Subproblem 1
z + -2 = 0.866025404 Simplifying z + -2 = 0.866025404 Reorder the terms: -2 + z = 0.866025404 Solving -2 + z = 0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 0.866025404 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.866025404 + 2 z = 0.866025404 + 2 Combine like terms: 0.866025404 + 2 = 2.866025404 z = 2.866025404 Simplifying z = 2.866025404Subproblem 2
z + -2 = -0.866025404 Simplifying z + -2 = -0.866025404 Reorder the terms: -2 + z = -0.866025404 Solving -2 + z = -0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -0.866025404 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.866025404 + 2 z = -0.866025404 + 2 Combine like terms: -0.866025404 + 2 = 1.133974596 z = 1.133974596 Simplifying z = 1.133974596Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.866025404, 1.133974596}
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