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Simplifying 8z + -3(5z2 + -3) + -1(4z2 + z + -5) = 0 Reorder the terms: 8z + -3(-3 + 5z2) + -1(4z2 + z + -5) = 0 8z + (-3 * -3 + 5z2 * -3) + -1(4z2 + z + -5) = 0 8z + (9 + -15z2) + -1(4z2 + z + -5) = 0 Reorder the terms: 8z + 9 + -15z2 + -1(-5 + z + 4z2) = 0 8z + 9 + -15z2 + (-5 * -1 + z * -1 + 4z2 * -1) = 0 8z + 9 + -15z2 + (5 + -1z + -4z2) = 0 Reorder the terms: 9 + 5 + 8z + -1z + -15z2 + -4z2 = 0 Combine like terms: 9 + 5 = 14 14 + 8z + -1z + -15z2 + -4z2 = 0 Combine like terms: 8z + -1z = 7z 14 + 7z + -15z2 + -4z2 = 0 Combine like terms: -15z2 + -4z2 = -19z2 14 + 7z + -19z2 = 0 Solving 14 + 7z + -19z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -19 the coefficient of the squared term: Divide each side by '-19'. -0.7368421053 + -0.3684210526z + z2 = 0 Move the constant term to the right: Add '0.7368421053' to each side of the equation. -0.7368421053 + -0.3684210526z + 0.7368421053 + z2 = 0 + 0.7368421053 Reorder the terms: -0.7368421053 + 0.7368421053 + -0.3684210526z + z2 = 0 + 0.7368421053 Combine like terms: -0.7368421053 + 0.7368421053 = 0.0000000000 0.0000000000 + -0.3684210526z + z2 = 0 + 0.7368421053 -0.3684210526z + z2 = 0 + 0.7368421053 Combine like terms: 0 + 0.7368421053 = 0.7368421053 -0.3684210526z + z2 = 0.7368421053 The z term is -0.3684210526z. Take half its coefficient (-0.1842105263). Square it (0.03393351800) and add it to both sides. Add '0.03393351800' to each side of the equation. -0.3684210526z + 0.03393351800 + z2 = 0.7368421053 + 0.03393351800 Reorder the terms: 0.03393351800 + -0.3684210526z + z2 = 0.7368421053 + 0.03393351800 Combine like terms: 0.7368421053 + 0.03393351800 = 0.7707756233 0.03393351800 + -0.3684210526z + z2 = 0.7707756233 Factor a perfect square on the left side: (z + -0.1842105263)(z + -0.1842105263) = 0.7707756233 Calculate the square root of the right side: 0.87793828 Break this problem into two subproblems by setting (z + -0.1842105263) equal to 0.87793828 and -0.87793828.Subproblem 1
z + -0.1842105263 = 0.87793828 Simplifying z + -0.1842105263 = 0.87793828 Reorder the terms: -0.1842105263 + z = 0.87793828 Solving -0.1842105263 + z = 0.87793828 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1842105263' to each side of the equation. -0.1842105263 + 0.1842105263 + z = 0.87793828 + 0.1842105263 Combine like terms: -0.1842105263 + 0.1842105263 = 0.0000000000 0.0000000000 + z = 0.87793828 + 0.1842105263 z = 0.87793828 + 0.1842105263 Combine like terms: 0.87793828 + 0.1842105263 = 1.0621488063 z = 1.0621488063 Simplifying z = 1.0621488063Subproblem 2
z + -0.1842105263 = -0.87793828 Simplifying z + -0.1842105263 = -0.87793828 Reorder the terms: -0.1842105263 + z = -0.87793828 Solving -0.1842105263 + z = -0.87793828 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1842105263' to each side of the equation. -0.1842105263 + 0.1842105263 + z = -0.87793828 + 0.1842105263 Combine like terms: -0.1842105263 + 0.1842105263 = 0.0000000000 0.0000000000 + z = -0.87793828 + 0.1842105263 z = -0.87793828 + 0.1842105263 Combine like terms: -0.87793828 + 0.1842105263 = -0.6937277537 z = -0.6937277537 Simplifying z = -0.6937277537Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.0621488063, -0.6937277537}
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