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Simplifying (7z2 + 8z) + -1(-3z2 + -7z + 3) = 0 Reorder the terms: (8z + 7z2) + -1(-3z2 + -7z + 3) = 0 Remove parenthesis around (8z + 7z2) 8z + 7z2 + -1(-3z2 + -7z + 3) = 0 Reorder the terms: 8z + 7z2 + -1(3 + -7z + -3z2) = 0 8z + 7z2 + (3 * -1 + -7z * -1 + -3z2 * -1) = 0 8z + 7z2 + (-3 + 7z + 3z2) = 0 Reorder the terms: -3 + 8z + 7z + 7z2 + 3z2 = 0 Combine like terms: 8z + 7z = 15z -3 + 15z + 7z2 + 3z2 = 0 Combine like terms: 7z2 + 3z2 = 10z2 -3 + 15z + 10z2 = 0 Solving -3 + 15z + 10z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.3 + 1.5z + z2 = 0.0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + 1.5z + 0.3 + z2 = 0.0 + 0.3 Reorder the terms: -0.3 + 0.3 + 1.5z + z2 = 0.0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + 1.5z + z2 = 0.0 + 0.3 1.5z + z2 = 0.0 + 0.3 Combine like terms: 0.0 + 0.3 = 0.3 1.5z + z2 = 0.3 The z term is 1.5z. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5z + 0.5625 + z2 = 0.3 + 0.5625 Reorder the terms: 0.5625 + 1.5z + z2 = 0.3 + 0.5625 Combine like terms: 0.3 + 0.5625 = 0.8625 0.5625 + 1.5z + z2 = 0.8625 Factor a perfect square on the left side: (z + 0.75)(z + 0.75) = 0.8625 Calculate the square root of the right side: 0.928708781 Break this problem into two subproblems by setting (z + 0.75) equal to 0.928708781 and -0.928708781.Subproblem 1
z + 0.75 = 0.928708781 Simplifying z + 0.75 = 0.928708781 Reorder the terms: 0.75 + z = 0.928708781 Solving 0.75 + z = 0.928708781 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = 0.928708781 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 0.928708781 + -0.75 z = 0.928708781 + -0.75 Combine like terms: 0.928708781 + -0.75 = 0.178708781 z = 0.178708781 Simplifying z = 0.178708781Subproblem 2
z + 0.75 = -0.928708781 Simplifying z + 0.75 = -0.928708781 Reorder the terms: 0.75 + z = -0.928708781 Solving 0.75 + z = -0.928708781 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = -0.928708781 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -0.928708781 + -0.75 z = -0.928708781 + -0.75 Combine like terms: -0.928708781 + -0.75 = -1.678708781 z = -1.678708781 Simplifying z = -1.678708781Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.178708781, -1.678708781}
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