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Simplifying (8z2 + -11z + 7) + (2z2 + 6z + -3) + -1(5z2 + 3z + 13) = 0 Reorder the terms: (7 + -11z + 8z2) + (2z2 + 6z + -3) + -1(5z2 + 3z + 13) = 0 Remove parenthesis around (7 + -11z + 8z2) 7 + -11z + 8z2 + (2z2 + 6z + -3) + -1(5z2 + 3z + 13) = 0 Reorder the terms: 7 + -11z + 8z2 + (-3 + 6z + 2z2) + -1(5z2 + 3z + 13) = 0 Remove parenthesis around (-3 + 6z + 2z2) 7 + -11z + 8z2 + -3 + 6z + 2z2 + -1(5z2 + 3z + 13) = 0 Reorder the terms: 7 + -11z + 8z2 + -3 + 6z + 2z2 + -1(13 + 3z + 5z2) = 0 7 + -11z + 8z2 + -3 + 6z + 2z2 + (13 * -1 + 3z * -1 + 5z2 * -1) = 0 7 + -11z + 8z2 + -3 + 6z + 2z2 + (-13 + -3z + -5z2) = 0 Reorder the terms: 7 + -3 + -13 + -11z + 6z + -3z + 8z2 + 2z2 + -5z2 = 0 Combine like terms: 7 + -3 = 4 4 + -13 + -11z + 6z + -3z + 8z2 + 2z2 + -5z2 = 0 Combine like terms: 4 + -13 = -9 -9 + -11z + 6z + -3z + 8z2 + 2z2 + -5z2 = 0 Combine like terms: -11z + 6z = -5z -9 + -5z + -3z + 8z2 + 2z2 + -5z2 = 0 Combine like terms: -5z + -3z = -8z -9 + -8z + 8z2 + 2z2 + -5z2 = 0 Combine like terms: 8z2 + 2z2 = 10z2 -9 + -8z + 10z2 + -5z2 = 0 Combine like terms: 10z2 + -5z2 = 5z2 -9 + -8z + 5z2 = 0 Solving -9 + -8z + 5z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.8 + -1.6z + z2 = 0 Move the constant term to the right: Add '1.8' to each side of the equation. -1.8 + -1.6z + 1.8 + z2 = 0 + 1.8 Reorder the terms: -1.8 + 1.8 + -1.6z + z2 = 0 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + -1.6z + z2 = 0 + 1.8 -1.6z + z2 = 0 + 1.8 Combine like terms: 0 + 1.8 = 1.8 -1.6z + z2 = 1.8 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 = 1.8 + 0.64 Reorder the terms: 0.64 + -1.6z + z2 = 1.8 + 0.64 Combine like terms: 1.8 + 0.64 = 2.44 0.64 + -1.6z + z2 = 2.44 Factor a perfect square on the left side: (z + -0.8)(z + -0.8) = 2.44 Calculate the square root of the right side: 1.562049935 Break this problem into two subproblems by setting (z + -0.8) equal to 1.562049935 and -1.562049935.Subproblem 1
z + -0.8 = 1.562049935 Simplifying z + -0.8 = 1.562049935 Reorder the terms: -0.8 + z = 1.562049935 Solving -0.8 + z = 1.562049935 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 = 1.562049935 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + z = 1.562049935 + 0.8 z = 1.562049935 + 0.8 Combine like terms: 1.562049935 + 0.8 = 2.362049935 z = 2.362049935 Simplifying z = 2.362049935Subproblem 2
z + -0.8 = -1.562049935 Simplifying z + -0.8 = -1.562049935 Reorder the terms: -0.8 + z = -1.562049935 Solving -0.8 + z = -1.562049935 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 = -1.562049935 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + z = -1.562049935 + 0.8 z = -1.562049935 + 0.8 Combine like terms: -1.562049935 + 0.8 = -0.762049935 z = -0.762049935 Simplifying z = -0.762049935Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.362049935, -0.762049935}
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