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Simplifying 7 + -3z + 8z2 + -8 + 4z = 0 Reorder the terms: 7 + -8 + -3z + 4z + 8z2 = 0 Combine like terms: 7 + -8 = -1 -1 + -3z + 4z + 8z2 = 0 Combine like terms: -3z + 4z = 1z -1 + 1z + 8z2 = 0 Solving -1 + 1z + 8z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + 0.125z + z2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 0.125z + 0.125 + z2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 0.125z + z2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 0.125z + z2 = 0 + 0.125 0.125z + z2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 0.125z + z2 = 0.125 The z term is 0.125z. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125z + 0.00390625 + z2 = 0.125 + 0.00390625 Reorder the terms: 0.00390625 + 0.125z + z2 = 0.125 + 0.00390625 Combine like terms: 0.125 + 0.00390625 = 0.12890625 0.00390625 + 0.125z + z2 = 0.12890625 Factor a perfect square on the left side: (z + 0.0625)(z + 0.0625) = 0.12890625 Calculate the square root of the right side: 0.359035165 Break this problem into two subproblems by setting (z + 0.0625) equal to 0.359035165 and -0.359035165.Subproblem 1
z + 0.0625 = 0.359035165 Simplifying z + 0.0625 = 0.359035165 Reorder the terms: 0.0625 + z = 0.359035165 Solving 0.0625 + z = 0.359035165 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + z = 0.359035165 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + z = 0.359035165 + -0.0625 z = 0.359035165 + -0.0625 Combine like terms: 0.359035165 + -0.0625 = 0.296535165 z = 0.296535165 Simplifying z = 0.296535165Subproblem 2
z + 0.0625 = -0.359035165 Simplifying z + 0.0625 = -0.359035165 Reorder the terms: 0.0625 + z = -0.359035165 Solving 0.0625 + z = -0.359035165 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + z = -0.359035165 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + z = -0.359035165 + -0.0625 z = -0.359035165 + -0.0625 Combine like terms: -0.359035165 + -0.0625 = -0.421535165 z = -0.421535165 Simplifying z = -0.421535165Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.296535165, -0.421535165}
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