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Simplifying -2z(5z + 8) + 13 = 0 Reorder the terms: -2z(8 + 5z) + 13 = 0 (8 * -2z + 5z * -2z) + 13 = 0 (-16z + -10z2) + 13 = 0 Reorder the terms: 13 + -16z + -10z2 = 0 Solving 13 + -16z + -10z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -1.3 + 1.6z + z2 = 0.0 Move the constant term to the right: Add '1.3' to each side of the equation. -1.3 + 1.6z + 1.3 + z2 = 0.0 + 1.3 Reorder the terms: -1.3 + 1.3 + 1.6z + z2 = 0.0 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + 1.6z + z2 = 0.0 + 1.3 1.6z + z2 = 0.0 + 1.3 Combine like terms: 0.0 + 1.3 = 1.3 1.6z + z2 = 1.3 The z term is 1.6z. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6z + 0.64 + z2 = 1.3 + 0.64 Reorder the terms: 0.64 + 1.6z + z2 = 1.3 + 0.64 Combine like terms: 1.3 + 0.64 = 1.94 0.64 + 1.6z + z2 = 1.94 Factor a perfect square on the left side: (z + 0.8)(z + 0.8) = 1.94 Calculate the square root of the right side: 1.392838828 Break this problem into two subproblems by setting (z + 0.8) equal to 1.392838828 and -1.392838828.Subproblem 1
z + 0.8 = 1.392838828 Simplifying z + 0.8 = 1.392838828 Reorder the terms: 0.8 + z = 1.392838828 Solving 0.8 + z = 1.392838828 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + z = 1.392838828 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = 1.392838828 + -0.8 z = 1.392838828 + -0.8 Combine like terms: 1.392838828 + -0.8 = 0.592838828 z = 0.592838828 Simplifying z = 0.592838828Subproblem 2
z + 0.8 = -1.392838828 Simplifying z + 0.8 = -1.392838828 Reorder the terms: 0.8 + z = -1.392838828 Solving 0.8 + z = -1.392838828 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + z = -1.392838828 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = -1.392838828 + -0.8 z = -1.392838828 + -0.8 Combine like terms: -1.392838828 + -0.8 = -2.192838828 z = -2.192838828 Simplifying z = -2.192838828Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.592838828, -2.192838828}
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