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Simplifying (5z2 + -11z + 2) + -1(12z2 + 9z + -11) = 0 Reorder the terms: (2 + -11z + 5z2) + -1(12z2 + 9z + -11) = 0 Remove parenthesis around (2 + -11z + 5z2) 2 + -11z + 5z2 + -1(12z2 + 9z + -11) = 0 Reorder the terms: 2 + -11z + 5z2 + -1(-11 + 9z + 12z2) = 0 2 + -11z + 5z2 + (-11 * -1 + 9z * -1 + 12z2 * -1) = 0 2 + -11z + 5z2 + (11 + -9z + -12z2) = 0 Reorder the terms: 2 + 11 + -11z + -9z + 5z2 + -12z2 = 0 Combine like terms: 2 + 11 = 13 13 + -11z + -9z + 5z2 + -12z2 = 0 Combine like terms: -11z + -9z = -20z 13 + -20z + 5z2 + -12z2 = 0 Combine like terms: 5z2 + -12z2 = -7z2 13 + -20z + -7z2 = 0 Solving 13 + -20z + -7z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.857142857 + 2.857142857z + z2 = 0 Move the constant term to the right: Add '1.857142857' to each side of the equation. -1.857142857 + 2.857142857z + 1.857142857 + z2 = 0 + 1.857142857 Reorder the terms: -1.857142857 + 1.857142857 + 2.857142857z + z2 = 0 + 1.857142857 Combine like terms: -1.857142857 + 1.857142857 = 0.000000000 0.000000000 + 2.857142857z + z2 = 0 + 1.857142857 2.857142857z + z2 = 0 + 1.857142857 Combine like terms: 0 + 1.857142857 = 1.857142857 2.857142857z + z2 = 1.857142857 The z term is 2.857142857z. Take half its coefficient (1.428571429). Square it (2.040816328) and add it to both sides. Add '2.040816328' to each side of the equation. 2.857142857z + 2.040816328 + z2 = 1.857142857 + 2.040816328 Reorder the terms: 2.040816328 + 2.857142857z + z2 = 1.857142857 + 2.040816328 Combine like terms: 1.857142857 + 2.040816328 = 3.897959185 2.040816328 + 2.857142857z + z2 = 3.897959185 Factor a perfect square on the left side: (z + 1.428571429)(z + 1.428571429) = 3.897959185 Calculate the square root of the right side: 1.974324995 Break this problem into two subproblems by setting (z + 1.428571429) equal to 1.974324995 and -1.974324995.Subproblem 1
z + 1.428571429 = 1.974324995 Simplifying z + 1.428571429 = 1.974324995 Reorder the terms: 1.428571429 + z = 1.974324995 Solving 1.428571429 + z = 1.974324995 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.428571429' to each side of the equation. 1.428571429 + -1.428571429 + z = 1.974324995 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + z = 1.974324995 + -1.428571429 z = 1.974324995 + -1.428571429 Combine like terms: 1.974324995 + -1.428571429 = 0.545753566 z = 0.545753566 Simplifying z = 0.545753566Subproblem 2
z + 1.428571429 = -1.974324995 Simplifying z + 1.428571429 = -1.974324995 Reorder the terms: 1.428571429 + z = -1.974324995 Solving 1.428571429 + z = -1.974324995 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.428571429' to each side of the equation. 1.428571429 + -1.428571429 + z = -1.974324995 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + z = -1.974324995 + -1.428571429 z = -1.974324995 + -1.428571429 Combine like terms: -1.974324995 + -1.428571429 = -3.402896424 z = -3.402896424 Simplifying z = -3.402896424Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.545753566, -3.402896424}
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