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Simplifying (7z2 + 8z) + -1(-9z2 + -9z + 4) = 0 Reorder the terms: (8z + 7z2) + -1(-9z2 + -9z + 4) = 0 Remove parenthesis around (8z + 7z2) 8z + 7z2 + -1(-9z2 + -9z + 4) = 0 Reorder the terms: 8z + 7z2 + -1(4 + -9z + -9z2) = 0 8z + 7z2 + (4 * -1 + -9z * -1 + -9z2 * -1) = 0 8z + 7z2 + (-4 + 9z + 9z2) = 0 Reorder the terms: -4 + 8z + 9z + 7z2 + 9z2 = 0 Combine like terms: 8z + 9z = 17z -4 + 17z + 7z2 + 9z2 = 0 Combine like terms: 7z2 + 9z2 = 16z2 -4 + 17z + 16z2 = 0 Solving -4 + 17z + 16z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.25 + 1.0625z + z2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 1.0625z + 0.25 + z2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 1.0625z + z2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 1.0625z + z2 = 0 + 0.25 1.0625z + z2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 1.0625z + z2 = 0.25 The z term is 1.0625z. Take half its coefficient (0.53125). Square it (0.2822265625) and add it to both sides. Add '0.2822265625' to each side of the equation. 1.0625z + 0.2822265625 + z2 = 0.25 + 0.2822265625 Reorder the terms: 0.2822265625 + 1.0625z + z2 = 0.25 + 0.2822265625 Combine like terms: 0.25 + 0.2822265625 = 0.5322265625 0.2822265625 + 1.0625z + z2 = 0.5322265625 Factor a perfect square on the left side: (z + 0.53125)(z + 0.53125) = 0.5322265625 Calculate the square root of the right side: 0.729538596 Break this problem into two subproblems by setting (z + 0.53125) equal to 0.729538596 and -0.729538596.Subproblem 1
z + 0.53125 = 0.729538596 Simplifying z + 0.53125 = 0.729538596 Reorder the terms: 0.53125 + z = 0.729538596 Solving 0.53125 + z = 0.729538596 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.53125' to each side of the equation. 0.53125 + -0.53125 + z = 0.729538596 + -0.53125 Combine like terms: 0.53125 + -0.53125 = 0.00000 0.00000 + z = 0.729538596 + -0.53125 z = 0.729538596 + -0.53125 Combine like terms: 0.729538596 + -0.53125 = 0.198288596 z = 0.198288596 Simplifying z = 0.198288596Subproblem 2
z + 0.53125 = -0.729538596 Simplifying z + 0.53125 = -0.729538596 Reorder the terms: 0.53125 + z = -0.729538596 Solving 0.53125 + z = -0.729538596 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.53125' to each side of the equation. 0.53125 + -0.53125 + z = -0.729538596 + -0.53125 Combine like terms: 0.53125 + -0.53125 = 0.00000 0.00000 + z = -0.729538596 + -0.53125 z = -0.729538596 + -0.53125 Combine like terms: -0.729538596 + -0.53125 = -1.260788596 z = -1.260788596 Simplifying z = -1.260788596Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.198288596, -1.260788596}
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