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Simplifying -3(3z + 3) * 3z = -53 Reorder the terms: -3(3 + 3z) * 3z = -53 Reorder the terms for easier multiplication: -3 * 3z(3 + 3z) = -53 Multiply -3 * 3 -9z(3 + 3z) = -53 (3 * -9z + 3z * -9z) = -53 (-27z + -27z2) = -53 Solving -27z + -27z2 = -53 Solving for variable 'z'. Reorder the terms: 53 + -27z + -27z2 = -53 + 53 Combine like terms: -53 + 53 = 0 53 + -27z + -27z2 = 0 Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. -1.962962963 + z + z2 = 0 Move the constant term to the right: Add '1.962962963' to each side of the equation. -1.962962963 + z + 1.962962963 + z2 = 0 + 1.962962963 Reorder the terms: -1.962962963 + 1.962962963 + z + z2 = 0 + 1.962962963 Combine like terms: -1.962962963 + 1.962962963 = 0.000000000 0.000000000 + z + z2 = 0 + 1.962962963 z + z2 = 0 + 1.962962963 Combine like terms: 0 + 1.962962963 = 1.962962963 z + z2 = 1.962962963 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + z2 = 1.962962963 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + z2 = 1.962962963 + 0.25 Combine like terms: 1.962962963 + 0.25 = 2.212962963 1.25 + z2 = 2.212962963 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 2.212962963 Calculate the square root of the right side: 1.487603093 Break this problem into two subproblems by setting (z + 0.5) equal to 1.487603093 and -1.487603093.Subproblem 1
z + 0.5 = 1.487603093 Simplifying z + 0.5 = 1.487603093 Reorder the terms: 0.5 + z = 1.487603093 Solving 0.5 + z = 1.487603093 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 1.487603093 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.487603093 + -0.5 z = 1.487603093 + -0.5 Combine like terms: 1.487603093 + -0.5 = 0.987603093 z = 0.987603093 Simplifying z = 0.987603093Subproblem 2
z + 0.5 = -1.487603093 Simplifying z + 0.5 = -1.487603093 Reorder the terms: 0.5 + z = -1.487603093 Solving 0.5 + z = -1.487603093 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -1.487603093 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.487603093 + -0.5 z = -1.487603093 + -0.5 Combine like terms: -1.487603093 + -0.5 = -1.987603093 z = -1.987603093 Simplifying z = -1.987603093Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.987603093, -1.987603093}
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