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Simplifying 10z(1 + -1z) = 0.9 (1 * 10z + -1z * 10z) = 0.9 (10z + -10z2) = 0.9 Solving 10z + -10z2 = 0.9 Solving for variable 'z'. Reorder the terms: -0.9 + 10z + -10z2 = 0.9 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 -0.9 + 10z + -10z2 = 0.0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. 0.09 + -1z + z2 = 0.00 Move the constant term to the right: Add '-0.09' to each side of the equation. 0.09 + -1z + -0.09 + z2 = 0.00 + -0.09 Reorder the terms: 0.09 + -0.09 + -1z + z2 = 0.00 + -0.09 Combine like terms: 0.09 + -0.09 = 0.00 0.00 + -1z + z2 = 0.00 + -0.09 -1z + z2 = 0.00 + -0.09 Combine like terms: 0.00 + -0.09 = -0.09 -1z + z2 = -0.09 The z term is -1z. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1z + 0.25 + z2 = -0.09 + 0.25 Reorder the terms: 0.25 + -1z + z2 = -0.09 + 0.25 Combine like terms: -0.09 + 0.25 = 0.16 0.25 + -1z + z2 = 0.16 Factor a perfect square on the left side: (z + -0.5)(z + -0.5) = 0.16 Calculate the square root of the right side: 0.4 Break this problem into two subproblems by setting (z + -0.5) equal to 0.4 and -0.4.Subproblem 1
z + -0.5 = 0.4 Simplifying z + -0.5 = 0.4 Reorder the terms: -0.5 + z = 0.4 Solving -0.5 + z = 0.4 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 0.4 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 0.4 + 0.5 z = 0.4 + 0.5 Combine like terms: 0.4 + 0.5 = 0.9 z = 0.9 Simplifying z = 0.9Subproblem 2
z + -0.5 = -0.4 Simplifying z + -0.5 = -0.4 Reorder the terms: -0.5 + z = -0.4 Solving -0.5 + z = -0.4 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -0.4 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -0.4 + 0.5 z = -0.4 + 0.5 Combine like terms: -0.4 + 0.5 = 0.1 z = 0.1 Simplifying z = 0.1Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.9, 0.1}
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