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Simplifying 3z2 + -11z + -20 = 20 Reorder the terms: -20 + -11z + 3z2 = 20 Solving -20 + -11z + 3z2 = 20 Solving for variable 'z'. Reorder the terms: -20 + -20 + -11z + 3z2 = 20 + -20 Combine like terms: -20 + -20 = -40 -40 + -11z + 3z2 = 20 + -20 Combine like terms: 20 + -20 = 0 -40 + -11z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -13.33333333 + -3.666666667z + z2 = 0 Move the constant term to the right: Add '13.33333333' to each side of the equation. -13.33333333 + -3.666666667z + 13.33333333 + z2 = 0 + 13.33333333 Reorder the terms: -13.33333333 + 13.33333333 + -3.666666667z + z2 = 0 + 13.33333333 Combine like terms: -13.33333333 + 13.33333333 = 0.00000000 0.00000000 + -3.666666667z + z2 = 0 + 13.33333333 -3.666666667z + z2 = 0 + 13.33333333 Combine like terms: 0 + 13.33333333 = 13.33333333 -3.666666667z + z2 = 13.33333333 The z term is -3.666666667z. Take half its coefficient (-1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. -3.666666667z + 3.361111114 + z2 = 13.33333333 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667z + z2 = 13.33333333 + 3.361111114 Combine like terms: 13.33333333 + 3.361111114 = 16.694444444 3.361111114 + -3.666666667z + z2 = 16.694444444 Factor a perfect square on the left side: (z + -1.833333334)(z + -1.833333334) = 16.694444444 Calculate the square root of the right side: 4.085883557 Break this problem into two subproblems by setting (z + -1.833333334) equal to 4.085883557 and -4.085883557.Subproblem 1
z + -1.833333334 = 4.085883557 Simplifying z + -1.833333334 = 4.085883557 Reorder the terms: -1.833333334 + z = 4.085883557 Solving -1.833333334 + z = 4.085883557 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + z = 4.085883557 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + z = 4.085883557 + 1.833333334 z = 4.085883557 + 1.833333334 Combine like terms: 4.085883557 + 1.833333334 = 5.919216891 z = 5.919216891 Simplifying z = 5.919216891Subproblem 2
z + -1.833333334 = -4.085883557 Simplifying z + -1.833333334 = -4.085883557 Reorder the terms: -1.833333334 + z = -4.085883557 Solving -1.833333334 + z = -4.085883557 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + z = -4.085883557 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + z = -4.085883557 + 1.833333334 z = -4.085883557 + 1.833333334 Combine like terms: -4.085883557 + 1.833333334 = -2.252550223 z = -2.252550223 Simplifying z = -2.252550223Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.919216891, -2.252550223}
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