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Simplifying 8z[-1(10z + 13) + 4] = 4z + 1 Reorder the terms: 8z[-1(13 + 10z) + 4] = 4z + 1 8z[(13 * -1 + 10z * -1) + 4] = 4z + 1 8z[(-13 + -10z) + 4] = 4z + 1 Reorder the terms: 8z[-13 + 4 + -10z] = 4z + 1 Combine like terms: -13 + 4 = -9 8z[-9 + -10z] = 4z + 1 [-9 * 8z + -10z * 8z] = 4z + 1 [-72z + -80z2] = 4z + 1 Reorder the terms: -72z + -80z2 = 1 + 4z Solving -72z + -80z2 = 1 + 4z Solving for variable 'z'. Reorder the terms: -1 + -72z + -4z + -80z2 = 1 + 4z + -1 + -4z Combine like terms: -72z + -4z = -76z -1 + -76z + -80z2 = 1 + 4z + -1 + -4z Reorder the terms: -1 + -76z + -80z2 = 1 + -1 + 4z + -4z Combine like terms: 1 + -1 = 0 -1 + -76z + -80z2 = 0 + 4z + -4z -1 + -76z + -80z2 = 4z + -4z Combine like terms: 4z + -4z = 0 -1 + -76z + -80z2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 76z + 80z2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 76z + 80z2)' equal to zero and attempt to solve: Simplifying 1 + 76z + 80z2 = 0 Solving 1 + 76z + 80z2 = 0 Begin completing the square. Divide all terms by 80 the coefficient of the squared term: Divide each side by '80'. 0.0125 + 0.95z + z2 = 0.0 Move the constant term to the right: Add '-0.0125' to each side of the equation. 0.0125 + 0.95z + -0.0125 + z2 = 0.0 + -0.0125 Reorder the terms: 0.0125 + -0.0125 + 0.95z + z2 = 0.0 + -0.0125 Combine like terms: 0.0125 + -0.0125 = 0.0000 0.0000 + 0.95z + z2 = 0.0 + -0.0125 0.95z + z2 = 0.0 + -0.0125 Combine like terms: 0.0 + -0.0125 = -0.0125 0.95z + z2 = -0.0125 The z term is 0.95z. Take half its coefficient (0.475). Square it (0.225625) and add it to both sides. Add '0.225625' to each side of the equation. 0.95z + 0.225625 + z2 = -0.0125 + 0.225625 Reorder the terms: 0.225625 + 0.95z + z2 = -0.0125 + 0.225625 Combine like terms: -0.0125 + 0.225625 = 0.213125 0.225625 + 0.95z + z2 = 0.213125 Factor a perfect square on the left side: (z + 0.475)(z + 0.475) = 0.213125 Calculate the square root of the right side: 0.461654633 Break this problem into two subproblems by setting (z + 0.475) equal to 0.461654633 and -0.461654633.Subproblem 1
z + 0.475 = 0.461654633 Simplifying z + 0.475 = 0.461654633 Reorder the terms: 0.475 + z = 0.461654633 Solving 0.475 + z = 0.461654633 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.475' to each side of the equation. 0.475 + -0.475 + z = 0.461654633 + -0.475 Combine like terms: 0.475 + -0.475 = 0.000 0.000 + z = 0.461654633 + -0.475 z = 0.461654633 + -0.475 Combine like terms: 0.461654633 + -0.475 = -0.013345367 z = -0.013345367 Simplifying z = -0.013345367Subproblem 2
z + 0.475 = -0.461654633 Simplifying z + 0.475 = -0.461654633 Reorder the terms: 0.475 + z = -0.461654633 Solving 0.475 + z = -0.461654633 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.475' to each side of the equation. 0.475 + -0.475 + z = -0.461654633 + -0.475 Combine like terms: 0.475 + -0.475 = 0.000 0.000 + z = -0.461654633 + -0.475 z = -0.461654633 + -0.475 Combine like terms: -0.461654633 + -0.475 = -0.936654633 z = -0.936654633 Simplifying z = -0.936654633Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.013345367, -0.936654633}Solution
z = {-0.013345367, -0.936654633}
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