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Simplifying 0.60z(z + -300) + 0.05z = 0.70z + -205 Reorder the terms: 0.60z(-300 + z) + 0.05z = 0.70z + -205 (-300 * 0.60z + z * 0.60z) + 0.05z = 0.70z + -205 (-180z + 0.60z2) + 0.05z = 0.70z + -205 Reorder the terms: -180z + 0.05z + 0.60z2 = 0.70z + -205 Combine like terms: -180z + 0.05z = -179.95z -179.95z + 0.60z2 = 0.70z + -205 Reorder the terms: -179.95z + 0.60z2 = -205 + 0.70z Solving -179.95z + 0.60z2 = -205 + 0.70z Solving for variable 'z'. Reorder the terms: 205 + -179.95z + -0.70z + 0.60z2 = -205 + 0.70z + 205 + -0.70z Combine like terms: -179.95z + -0.70z = -180.65z 205 + -180.65z + 0.60z2 = -205 + 0.70z + 205 + -0.70z Reorder the terms: 205 + -180.65z + 0.60z2 = -205 + 205 + 0.70z + -0.70z Combine like terms: -205 + 205 = 0 205 + -180.65z + 0.60z2 = 0 + 0.70z + -0.70z 205 + -180.65z + 0.60z2 = 0.70z + -0.70z Combine like terms: 0.70z + -0.70z = 0.00 205 + -180.65z + 0.60z2 = 0.00 Begin completing the square. Divide all terms by 0.60 the coefficient of the squared term: Divide each side by '0.60'. 341.6666667 + -301.0833333z + z2 = 0 Move the constant term to the right: Add '-341.6666667' to each side of the equation. 341.6666667 + -301.0833333z + -341.6666667 + z2 = 0 + -341.6666667 Reorder the terms: 341.6666667 + -341.6666667 + -301.0833333z + z2 = 0 + -341.6666667 Combine like terms: 341.6666667 + -341.6666667 = 0.0000000 0.0000000 + -301.0833333z + z2 = 0 + -341.6666667 -301.0833333z + z2 = 0 + -341.6666667 Combine like terms: 0 + -341.6666667 = -341.6666667 -301.0833333z + z2 = -341.6666667 The z term is -301.0833333z. Take half its coefficient (-150.5416667). Square it (22662.79341) and add it to both sides. Add '22662.79341' to each side of the equation. -301.0833333z + 22662.79341 + z2 = -341.6666667 + 22662.79341 Reorder the terms: 22662.79341 + -301.0833333z + z2 = -341.6666667 + 22662.79341 Combine like terms: -341.6666667 + 22662.79341 = 22321.1267433 22662.79341 + -301.0833333z + z2 = 22321.1267433 Factor a perfect square on the left side: (z + -150.5416667)(z + -150.5416667) = 22321.1267433 Calculate the square root of the right side: 149.402566053 Break this problem into two subproblems by setting (z + -150.5416667) equal to 149.402566053 and -149.402566053.Subproblem 1
z + -150.5416667 = 149.402566053 Simplifying z + -150.5416667 = 149.402566053 Reorder the terms: -150.5416667 + z = 149.402566053 Solving -150.5416667 + z = 149.402566053 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '150.5416667' to each side of the equation. -150.5416667 + 150.5416667 + z = 149.402566053 + 150.5416667 Combine like terms: -150.5416667 + 150.5416667 = 0.0000000 0.0000000 + z = 149.402566053 + 150.5416667 z = 149.402566053 + 150.5416667 Combine like terms: 149.402566053 + 150.5416667 = 299.944232753 z = 299.944232753 Simplifying z = 299.944232753Subproblem 2
z + -150.5416667 = -149.402566053 Simplifying z + -150.5416667 = -149.402566053 Reorder the terms: -150.5416667 + z = -149.402566053 Solving -150.5416667 + z = -149.402566053 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '150.5416667' to each side of the equation. -150.5416667 + 150.5416667 + z = -149.402566053 + 150.5416667 Combine like terms: -150.5416667 + 150.5416667 = 0.0000000 0.0000000 + z = -149.402566053 + 150.5416667 z = -149.402566053 + 150.5416667 Combine like terms: -149.402566053 + 150.5416667 = 1.139100647 z = 1.139100647 Simplifying z = 1.139100647Solution
The solution to the problem is based on the solutions from the subproblems. z = {299.944232753, 1.139100647}
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