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Simplifying 8z(8z) = 6z + 38 Remove parenthesis around (8z) 8z * 8z = 6z + 38 Reorder the terms for easier multiplication: 8 * 8z * z = 6z + 38 Multiply 8 * 8 64z * z = 6z + 38 Multiply z * z 64z2 = 6z + 38 Reorder the terms: 64z2 = 38 + 6z Solving 64z2 = 38 + 6z Solving for variable 'z'. Reorder the terms: -38 + -6z + 64z2 = 38 + 6z + -38 + -6z Reorder the terms: -38 + -6z + 64z2 = 38 + -38 + 6z + -6z Combine like terms: 38 + -38 = 0 -38 + -6z + 64z2 = 0 + 6z + -6z -38 + -6z + 64z2 = 6z + -6z Combine like terms: 6z + -6z = 0 -38 + -6z + 64z2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-19 + -3z + 32z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-19 + -3z + 32z2)' equal to zero and attempt to solve: Simplifying -19 + -3z + 32z2 = 0 Solving -19 + -3z + 32z2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.59375 + -0.09375z + z2 = 0 Move the constant term to the right: Add '0.59375' to each side of the equation. -0.59375 + -0.09375z + 0.59375 + z2 = 0 + 0.59375 Reorder the terms: -0.59375 + 0.59375 + -0.09375z + z2 = 0 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + -0.09375z + z2 = 0 + 0.59375 -0.09375z + z2 = 0 + 0.59375 Combine like terms: 0 + 0.59375 = 0.59375 -0.09375z + z2 = 0.59375 The z term is -0.09375z. Take half its coefficient (-0.046875). Square it (0.002197265625) and add it to both sides. Add '0.002197265625' to each side of the equation. -0.09375z + 0.002197265625 + z2 = 0.59375 + 0.002197265625 Reorder the terms: 0.002197265625 + -0.09375z + z2 = 0.59375 + 0.002197265625 Combine like terms: 0.59375 + 0.002197265625 = 0.595947265625 0.002197265625 + -0.09375z + z2 = 0.595947265625 Factor a perfect square on the left side: (z + -0.046875)(z + -0.046875) = 0.595947265625 Calculate the square root of the right side: 0.771976208 Break this problem into two subproblems by setting (z + -0.046875) equal to 0.771976208 and -0.771976208.Subproblem 1
z + -0.046875 = 0.771976208 Simplifying z + -0.046875 = 0.771976208 Reorder the terms: -0.046875 + z = 0.771976208 Solving -0.046875 + z = 0.771976208 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.046875' to each side of the equation. -0.046875 + 0.046875 + z = 0.771976208 + 0.046875 Combine like terms: -0.046875 + 0.046875 = 0.000000 0.000000 + z = 0.771976208 + 0.046875 z = 0.771976208 + 0.046875 Combine like terms: 0.771976208 + 0.046875 = 0.818851208 z = 0.818851208 Simplifying z = 0.818851208Subproblem 2
z + -0.046875 = -0.771976208 Simplifying z + -0.046875 = -0.771976208 Reorder the terms: -0.046875 + z = -0.771976208 Solving -0.046875 + z = -0.771976208 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.046875' to each side of the equation. -0.046875 + 0.046875 + z = -0.771976208 + 0.046875 Combine like terms: -0.046875 + 0.046875 = 0.000000 0.000000 + z = -0.771976208 + 0.046875 z = -0.771976208 + 0.046875 Combine like terms: -0.771976208 + 0.046875 = -0.725101208 z = -0.725101208 Simplifying z = -0.725101208Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.818851208, -0.725101208}Solution
z = {0.818851208, -0.725101208}
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