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Simplifying 4z2 + -1z + -0.5 = 0 Reorder the terms: -0.5 + -1z + 4z2 = 0 Solving -0.5 + -1z + 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'. -0.125 + -0.25z + z2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -0.25z + 0.125 + z2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -0.25z + z2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -0.25z + z2 = 0 + 0.125 -0.25z + z2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -0.25z + z2 = 0.125 The z term is -0.25z. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25z + 0.015625 + z2 = 0.125 + 0.015625 Reorder the terms: 0.015625 + -0.25z + z2 = 0.125 + 0.015625 Combine like terms: 0.125 + 0.015625 = 0.140625 0.015625 + -0.25z + z2 = 0.140625 Factor a perfect square on the left side: (z + -0.125)(z + -0.125) = 0.140625 Calculate the square root of the right side: 0.375 Break this problem into two subproblems by setting (z + -0.125) equal to 0.375 and -0.375.Subproblem 1
z + -0.125 = 0.375 Simplifying z + -0.125 = 0.375 Reorder the terms: -0.125 + z = 0.375 Solving -0.125 + z = 0.375 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + z = 0.375 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = 0.375 + 0.125 z = 0.375 + 0.125 Combine like terms: 0.375 + 0.125 = 0.5 z = 0.5 Simplifying z = 0.5Subproblem 2
z + -0.125 = -0.375 Simplifying z + -0.125 = -0.375 Reorder the terms: -0.125 + z = -0.375 Solving -0.125 + z = -0.375 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + z = -0.375 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = -0.375 + 0.125 z = -0.375 + 0.125 Combine like terms: -0.375 + 0.125 = -0.25 z = -0.25 Simplifying z = -0.25Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.5, -0.25}
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