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Simplifying 4z + -1 = 25z2 + -14z + 2 Reorder the terms: -1 + 4z = 25z2 + -14z + 2 Reorder the terms: -1 + 4z = 2 + -14z + 25z2 Solving -1 + 4z = 2 + -14z + 25z2 Solving for variable 'z'. Reorder the terms: -1 + -2 + 4z + 14z + -25z2 = 2 + -14z + 25z2 + -2 + 14z + -25z2 Combine like terms: -1 + -2 = -3 -3 + 4z + 14z + -25z2 = 2 + -14z + 25z2 + -2 + 14z + -25z2 Combine like terms: 4z + 14z = 18z -3 + 18z + -25z2 = 2 + -14z + 25z2 + -2 + 14z + -25z2 Reorder the terms: -3 + 18z + -25z2 = 2 + -2 + -14z + 14z + 25z2 + -25z2 Combine like terms: 2 + -2 = 0 -3 + 18z + -25z2 = 0 + -14z + 14z + 25z2 + -25z2 -3 + 18z + -25z2 = -14z + 14z + 25z2 + -25z2 Combine like terms: -14z + 14z = 0 -3 + 18z + -25z2 = 0 + 25z2 + -25z2 -3 + 18z + -25z2 = 25z2 + -25z2 Combine like terms: 25z2 + -25z2 = 0 -3 + 18z + -25z2 = 0 Begin completing the square. Divide all terms by -25 the coefficient of the squared term: Divide each side by '-25'. 0.12 + -0.72z + z2 = 0 Move the constant term to the right: Add '-0.12' to each side of the equation. 0.12 + -0.72z + -0.12 + z2 = 0 + -0.12 Reorder the terms: 0.12 + -0.12 + -0.72z + z2 = 0 + -0.12 Combine like terms: 0.12 + -0.12 = 0.00 0.00 + -0.72z + z2 = 0 + -0.12 -0.72z + z2 = 0 + -0.12 Combine like terms: 0 + -0.12 = -0.12 -0.72z + z2 = -0.12 The z term is -0.72z. Take half its coefficient (-0.36). Square it (0.1296) and add it to both sides. Add '0.1296' to each side of the equation. -0.72z + 0.1296 + z2 = -0.12 + 0.1296 Reorder the terms: 0.1296 + -0.72z + z2 = -0.12 + 0.1296 Combine like terms: -0.12 + 0.1296 = 0.0096 0.1296 + -0.72z + z2 = 0.0096 Factor a perfect square on the left side: (z + -0.36)(z + -0.36) = 0.0096 Calculate the square root of the right side: 0.09797959 Break this problem into two subproblems by setting (z + -0.36) equal to 0.09797959 and -0.09797959.Subproblem 1
z + -0.36 = 0.09797959 Simplifying z + -0.36 = 0.09797959 Reorder the terms: -0.36 + z = 0.09797959 Solving -0.36 + z = 0.09797959 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.36' to each side of the equation. -0.36 + 0.36 + z = 0.09797959 + 0.36 Combine like terms: -0.36 + 0.36 = 0.00 0.00 + z = 0.09797959 + 0.36 z = 0.09797959 + 0.36 Combine like terms: 0.09797959 + 0.36 = 0.45797959 z = 0.45797959 Simplifying z = 0.45797959Subproblem 2
z + -0.36 = -0.09797959 Simplifying z + -0.36 = -0.09797959 Reorder the terms: -0.36 + z = -0.09797959 Solving -0.36 + z = -0.09797959 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.36' to each side of the equation. -0.36 + 0.36 + z = -0.09797959 + 0.36 Combine like terms: -0.36 + 0.36 = 0.00 0.00 + z = -0.09797959 + 0.36 z = -0.09797959 + 0.36 Combine like terms: -0.09797959 + 0.36 = 0.26202041 z = 0.26202041 Simplifying z = 0.26202041Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.45797959, 0.26202041}
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