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Simplifying 3(2z + 7) + 2(z + 4) = 5(z + -6) * 9z + 5 Reorder the terms: 3(7 + 2z) + 2(z + 4) = 5(z + -6) * 9z + 5 (7 * 3 + 2z * 3) + 2(z + 4) = 5(z + -6) * 9z + 5 (21 + 6z) + 2(z + 4) = 5(z + -6) * 9z + 5 Reorder the terms: 21 + 6z + 2(4 + z) = 5(z + -6) * 9z + 5 21 + 6z + (4 * 2 + z * 2) = 5(z + -6) * 9z + 5 21 + 6z + (8 + 2z) = 5(z + -6) * 9z + 5 Reorder the terms: 21 + 8 + 6z + 2z = 5(z + -6) * 9z + 5 Combine like terms: 21 + 8 = 29 29 + 6z + 2z = 5(z + -6) * 9z + 5 Combine like terms: 6z + 2z = 8z 29 + 8z = 5(z + -6) * 9z + 5 Reorder the terms: 29 + 8z = 5(-6 + z) * 9z + 5 Reorder the terms for easier multiplication: 29 + 8z = 5 * 9z(-6 + z) + 5 Multiply 5 * 9 29 + 8z = 45z(-6 + z) + 5 29 + 8z = (-6 * 45z + z * 45z) + 5 29 + 8z = (-270z + 45z2) + 5 Reorder the terms: 29 + 8z = 5 + -270z + 45z2 Solving 29 + 8z = 5 + -270z + 45z2 Solving for variable 'z'. Reorder the terms: 29 + -5 + 8z + 270z + -45z2 = 5 + -270z + 45z2 + -5 + 270z + -45z2 Combine like terms: 29 + -5 = 24 24 + 8z + 270z + -45z2 = 5 + -270z + 45z2 + -5 + 270z + -45z2 Combine like terms: 8z + 270z = 278z 24 + 278z + -45z2 = 5 + -270z + 45z2 + -5 + 270z + -45z2 Reorder the terms: 24 + 278z + -45z2 = 5 + -5 + -270z + 270z + 45z2 + -45z2 Combine like terms: 5 + -5 = 0 24 + 278z + -45z2 = 0 + -270z + 270z + 45z2 + -45z2 24 + 278z + -45z2 = -270z + 270z + 45z2 + -45z2 Combine like terms: -270z + 270z = 0 24 + 278z + -45z2 = 0 + 45z2 + -45z2 24 + 278z + -45z2 = 45z2 + -45z2 Combine like terms: 45z2 + -45z2 = 0 24 + 278z + -45z2 = 0 Begin completing the square. Divide all terms by -45 the coefficient of the squared term: Divide each side by '-45'. -0.5333333333 + -6.177777778z + z2 = 0 Move the constant term to the right: Add '0.5333333333' to each side of the equation. -0.5333333333 + -6.177777778z + 0.5333333333 + z2 = 0 + 0.5333333333 Reorder the terms: -0.5333333333 + 0.5333333333 + -6.177777778z + z2 = 0 + 0.5333333333 Combine like terms: -0.5333333333 + 0.5333333333 = 0.0000000000 0.0000000000 + -6.177777778z + z2 = 0 + 0.5333333333 -6.177777778z + z2 = 0 + 0.5333333333 Combine like terms: 0 + 0.5333333333 = 0.5333333333 -6.177777778z + z2 = 0.5333333333 The z term is -6.177777778z. Take half its coefficient (-3.088888889). Square it (9.541234569) and add it to both sides. Add '9.541234569' to each side of the equation. -6.177777778z + 9.541234569 + z2 = 0.5333333333 + 9.541234569 Reorder the terms: 9.541234569 + -6.177777778z + z2 = 0.5333333333 + 9.541234569 Combine like terms: 0.5333333333 + 9.541234569 = 10.0745679023 9.541234569 + -6.177777778z + z2 = 10.0745679023 Factor a perfect square on the left side: (z + -3.088888889)(z + -3.088888889) = 10.0745679023 Calculate the square root of the right side: 3.174045983 Break this problem into two subproblems by setting (z + -3.088888889) equal to 3.174045983 and -3.174045983.Subproblem 1
z + -3.088888889 = 3.174045983 Simplifying z + -3.088888889 = 3.174045983 Reorder the terms: -3.088888889 + z = 3.174045983 Solving -3.088888889 + z = 3.174045983 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.088888889' to each side of the equation. -3.088888889 + 3.088888889 + z = 3.174045983 + 3.088888889 Combine like terms: -3.088888889 + 3.088888889 = 0.000000000 0.000000000 + z = 3.174045983 + 3.088888889 z = 3.174045983 + 3.088888889 Combine like terms: 3.174045983 + 3.088888889 = 6.262934872 z = 6.262934872 Simplifying z = 6.262934872Subproblem 2
z + -3.088888889 = -3.174045983 Simplifying z + -3.088888889 = -3.174045983 Reorder the terms: -3.088888889 + z = -3.174045983 Solving -3.088888889 + z = -3.174045983 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.088888889' to each side of the equation. -3.088888889 + 3.088888889 + z = -3.174045983 + 3.088888889 Combine like terms: -3.088888889 + 3.088888889 = 0.000000000 0.000000000 + z = -3.174045983 + 3.088888889 z = -3.174045983 + 3.088888889 Combine like terms: -3.174045983 + 3.088888889 = -0.085157094 z = -0.085157094 Simplifying z = -0.085157094Solution
The solution to the problem is based on the solutions from the subproblems. z = {6.262934872, -0.085157094}
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