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Simplifying 0 = 100 + -75z + -50z2 Solving 0 = 100 + -75z + -50z2 Solving for variable 'z'. Combine like terms: 0 + -100 = -100 -100 + 75z + 50z2 = 100 + -75z + -50z2 + -100 + 75z + 50z2 Reorder the terms: -100 + 75z + 50z2 = 100 + -100 + -75z + 75z + -50z2 + 50z2 Combine like terms: 100 + -100 = 0 -100 + 75z + 50z2 = 0 + -75z + 75z + -50z2 + 50z2 -100 + 75z + 50z2 = -75z + 75z + -50z2 + 50z2 Combine like terms: -75z + 75z = 0 -100 + 75z + 50z2 = 0 + -50z2 + 50z2 -100 + 75z + 50z2 = -50z2 + 50z2 Combine like terms: -50z2 + 50z2 = 0 -100 + 75z + 50z2 = 0 Factor out the Greatest Common Factor (GCF), '25'. 25(-4 + 3z + 2z2) = 0 Ignore the factor 25.Subproblem 1
Set the factor '(-4 + 3z + 2z2)' equal to zero and attempt to solve: Simplifying -4 + 3z + 2z2 = 0 Solving -4 + 3z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + 1.5z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.5z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 1.5z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.5z + z2 = 0 + 2 1.5z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.5z + z2 = 2 The z term is 1.5z. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5z + 0.5625 + z2 = 2 + 0.5625 Reorder the terms: 0.5625 + 1.5z + z2 = 2 + 0.5625 Combine like terms: 2 + 0.5625 = 2.5625 0.5625 + 1.5z + z2 = 2.5625 Factor a perfect square on the left side: (z + 0.75)(z + 0.75) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (z + 0.75) equal to 1.600781059 and -1.600781059.Subproblem 1
z + 0.75 = 1.600781059 Simplifying z + 0.75 = 1.600781059 Reorder the terms: 0.75 + z = 1.600781059 Solving 0.75 + z = 1.600781059 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = 1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.600781059 + -0.75 z = 1.600781059 + -0.75 Combine like terms: 1.600781059 + -0.75 = 0.850781059 z = 0.850781059 Simplifying z = 0.850781059Subproblem 2
z + 0.75 = -1.600781059 Simplifying z + 0.75 = -1.600781059 Reorder the terms: 0.75 + z = -1.600781059 Solving 0.75 + z = -1.600781059 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = -1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.600781059 + -0.75 z = -1.600781059 + -0.75 Combine like terms: -1.600781059 + -0.75 = -2.350781059 z = -2.350781059 Simplifying z = -2.350781059Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.850781059, -2.350781059}Solution
z = {0.850781059, -2.350781059}
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