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Simplifying (3z2 + 5z + -1) + -1(7z2 + -7z + 8) + (5z2 + -6z + 1) = 0 Reorder the terms: (-1 + 5z + 3z2) + -1(7z2 + -7z + 8) + (5z2 + -6z + 1) = 0 Remove parenthesis around (-1 + 5z + 3z2) -1 + 5z + 3z2 + -1(7z2 + -7z + 8) + (5z2 + -6z + 1) = 0 Reorder the terms: -1 + 5z + 3z2 + -1(8 + -7z + 7z2) + (5z2 + -6z + 1) = 0 -1 + 5z + 3z2 + (8 * -1 + -7z * -1 + 7z2 * -1) + (5z2 + -6z + 1) = 0 -1 + 5z + 3z2 + (-8 + 7z + -7z2) + (5z2 + -6z + 1) = 0 Reorder the terms: -1 + 5z + 3z2 + -8 + 7z + -7z2 + (1 + -6z + 5z2) = 0 Remove parenthesis around (1 + -6z + 5z2) -1 + 5z + 3z2 + -8 + 7z + -7z2 + 1 + -6z + 5z2 = 0 Reorder the terms: -1 + -8 + 1 + 5z + 7z + -6z + 3z2 + -7z2 + 5z2 = 0 Combine like terms: -1 + -8 = -9 -9 + 1 + 5z + 7z + -6z + 3z2 + -7z2 + 5z2 = 0 Combine like terms: -9 + 1 = -8 -8 + 5z + 7z + -6z + 3z2 + -7z2 + 5z2 = 0 Combine like terms: 5z + 7z = 12z -8 + 12z + -6z + 3z2 + -7z2 + 5z2 = 0 Combine like terms: 12z + -6z = 6z -8 + 6z + 3z2 + -7z2 + 5z2 = 0 Combine like terms: 3z2 + -7z2 = -4z2 -8 + 6z + -4z2 + 5z2 = 0 Combine like terms: -4z2 + 5z2 = 1z2 -8 + 6z + 1z2 = 0 Solving -8 + 6z + 1z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '8' to each side of the equation. -8 + 6z + 8 + z2 = 0 + 8 Reorder the terms: -8 + 8 + 6z + z2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 6z + z2 = 0 + 8 6z + z2 = 0 + 8 Combine like terms: 0 + 8 = 8 6z + z2 = 8 The z term is 6z. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6z + 9 + z2 = 8 + 9 Reorder the terms: 9 + 6z + z2 = 8 + 9 Combine like terms: 8 + 9 = 17 9 + 6z + z2 = 17 Factor a perfect square on the left side: (z + 3)(z + 3) = 17 Calculate the square root of the right side: 4.123105626 Break this problem into two subproblems by setting (z + 3) equal to 4.123105626 and -4.123105626.Subproblem 1
z + 3 = 4.123105626 Simplifying z + 3 = 4.123105626 Reorder the terms: 3 + z = 4.123105626 Solving 3 + z = 4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = 4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + z = 4.123105626 + -3 z = 4.123105626 + -3 Combine like terms: 4.123105626 + -3 = 1.123105626 z = 1.123105626 Simplifying z = 1.123105626Subproblem 2
z + 3 = -4.123105626 Simplifying z + 3 = -4.123105626 Reorder the terms: 3 + z = -4.123105626 Solving 3 + z = -4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = -4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + z = -4.123105626 + -3 z = -4.123105626 + -3 Combine like terms: -4.123105626 + -3 = -7.123105626 z = -7.123105626 Simplifying z = -7.123105626Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.123105626, -7.123105626}
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