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Simplifying 4z2 + -1z = 54 Reorder the terms: -1z + 4z2 = 54 Solving -1z + 4z2 = 54 Solving for variable 'z'. Reorder the terms: -54 + -1z + 4z2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + -1z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -13.5 + -0.25z + z2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + -0.25z + 13.5 + z2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + -0.25z + z2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + -0.25z + z2 = 0 + 13.5 -0.25z + z2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 -0.25z + z2 = 13.5 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 = 13.5 + 0.015625 Reorder the terms: 0.015625 + -0.25z + z2 = 13.5 + 0.015625 Combine like terms: 13.5 + 0.015625 = 13.515625 0.015625 + -0.25z + z2 = 13.515625 Factor a perfect square on the left side: (z + -0.125)(z + -0.125) = 13.515625 Calculate the square root of the right side: 3.676360292 Break this problem into two subproblems by setting (z + -0.125) equal to 3.676360292 and -3.676360292.Subproblem 1
z + -0.125 = 3.676360292 Simplifying z + -0.125 = 3.676360292 Reorder the terms: -0.125 + z = 3.676360292 Solving -0.125 + z = 3.676360292 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 = 3.676360292 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = 3.676360292 + 0.125 z = 3.676360292 + 0.125 Combine like terms: 3.676360292 + 0.125 = 3.801360292 z = 3.801360292 Simplifying z = 3.801360292Subproblem 2
z + -0.125 = -3.676360292 Simplifying z + -0.125 = -3.676360292 Reorder the terms: -0.125 + z = -3.676360292 Solving -0.125 + z = -3.676360292 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 = -3.676360292 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = -3.676360292 + 0.125 z = -3.676360292 + 0.125 Combine like terms: -3.676360292 + 0.125 = -3.551360292 z = -3.551360292 Simplifying z = -3.551360292Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.801360292, -3.551360292}
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