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Simplifying -4 + 6(5z + 3) = -1z(z + 4) + -5 Reorder the terms: -4 + 6(3 + 5z) = -1z(z + 4) + -5 -4 + (3 * 6 + 5z * 6) = -1z(z + 4) + -5 -4 + (18 + 30z) = -1z(z + 4) + -5 Combine like terms: -4 + 18 = 14 14 + 30z = -1z(z + 4) + -5 Reorder the terms: 14 + 30z = -1z(4 + z) + -5 14 + 30z = (4 * -1z + z * -1z) + -5 14 + 30z = (-4z + -1z2) + -5 Reorder the terms: 14 + 30z = -5 + -4z + -1z2 Solving 14 + 30z = -5 + -4z + -1z2 Solving for variable 'z'. Reorder the terms: 14 + 5 + 30z + 4z + z2 = -5 + -4z + -1z2 + 5 + 4z + z2 Combine like terms: 14 + 5 = 19 19 + 30z + 4z + z2 = -5 + -4z + -1z2 + 5 + 4z + z2 Combine like terms: 30z + 4z = 34z 19 + 34z + z2 = -5 + -4z + -1z2 + 5 + 4z + z2 Reorder the terms: 19 + 34z + z2 = -5 + 5 + -4z + 4z + -1z2 + z2 Combine like terms: -5 + 5 = 0 19 + 34z + z2 = 0 + -4z + 4z + -1z2 + z2 19 + 34z + z2 = -4z + 4z + -1z2 + z2 Combine like terms: -4z + 4z = 0 19 + 34z + z2 = 0 + -1z2 + z2 19 + 34z + z2 = -1z2 + z2 Combine like terms: -1z2 + z2 = 0 19 + 34z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + 34z + -19 + z2 = 0 + -19 Reorder the terms: 19 + -19 + 34z + z2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + 34z + z2 = 0 + -19 34z + z2 = 0 + -19 Combine like terms: 0 + -19 = -19 34z + z2 = -19 The z term is 34z. Take half its coefficient (17). Square it (289) and add it to both sides. Add '289' to each side of the equation. 34z + 289 + z2 = -19 + 289 Reorder the terms: 289 + 34z + z2 = -19 + 289 Combine like terms: -19 + 289 = 270 289 + 34z + z2 = 270 Factor a perfect square on the left side: (z + 17)(z + 17) = 270 Calculate the square root of the right side: 16.431676725 Break this problem into two subproblems by setting (z + 17) equal to 16.431676725 and -16.431676725.Subproblem 1
z + 17 = 16.431676725 Simplifying z + 17 = 16.431676725 Reorder the terms: 17 + z = 16.431676725 Solving 17 + z = 16.431676725 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-17' to each side of the equation. 17 + -17 + z = 16.431676725 + -17 Combine like terms: 17 + -17 = 0 0 + z = 16.431676725 + -17 z = 16.431676725 + -17 Combine like terms: 16.431676725 + -17 = -0.568323275 z = -0.568323275 Simplifying z = -0.568323275Subproblem 2
z + 17 = -16.431676725 Simplifying z + 17 = -16.431676725 Reorder the terms: 17 + z = -16.431676725 Solving 17 + z = -16.431676725 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-17' to each side of the equation. 17 + -17 + z = -16.431676725 + -17 Combine like terms: 17 + -17 = 0 0 + z = -16.431676725 + -17 z = -16.431676725 + -17 Combine like terms: -16.431676725 + -17 = -33.431676725 z = -33.431676725 Simplifying z = -33.431676725Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.568323275, -33.431676725}
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