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Simplifying -1z2 + -7z + 34 = 0 Reorder the terms: 34 + -7z + -1z2 = 0 Solving 34 + -7z + -1z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -34 + 7z + z2 = 0 Move the constant term to the right: Add '34' to each side of the equation. -34 + 7z + 34 + z2 = 0 + 34 Reorder the terms: -34 + 34 + 7z + z2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 7z + z2 = 0 + 34 7z + z2 = 0 + 34 Combine like terms: 0 + 34 = 34 7z + z2 = 34 The z term is 7z. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7z + 12.25 + z2 = 34 + 12.25 Reorder the terms: 12.25 + 7z + z2 = 34 + 12.25 Combine like terms: 34 + 12.25 = 46.25 12.25 + 7z + z2 = 46.25 Factor a perfect square on the left side: (z + 3.5)(z + 3.5) = 46.25 Calculate the square root of the right side: 6.800735254 Break this problem into two subproblems by setting (z + 3.5) equal to 6.800735254 and -6.800735254.Subproblem 1
z + 3.5 = 6.800735254 Simplifying z + 3.5 = 6.800735254 Reorder the terms: 3.5 + z = 6.800735254 Solving 3.5 + z = 6.800735254 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 6.800735254 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 6.800735254 + -3.5 z = 6.800735254 + -3.5 Combine like terms: 6.800735254 + -3.5 = 3.300735254 z = 3.300735254 Simplifying z = 3.300735254Subproblem 2
z + 3.5 = -6.800735254 Simplifying z + 3.5 = -6.800735254 Reorder the terms: 3.5 + z = -6.800735254 Solving 3.5 + z = -6.800735254 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -6.800735254 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -6.800735254 + -3.5 z = -6.800735254 + -3.5 Combine like terms: -6.800735254 + -3.5 = -10.300735254 z = -10.300735254 Simplifying z = -10.300735254Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.300735254, -10.300735254}
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