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Simplifying -1z2 + -7z + 20 = 0 Reorder the terms: 20 + -7z + -1z2 = 0 Solving 20 + -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'. -20 + 7z + z2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + 7z + 20 + z2 = 0 + 20 Reorder the terms: -20 + 20 + 7z + z2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 7z + z2 = 0 + 20 7z + z2 = 0 + 20 Combine like terms: 0 + 20 = 20 7z + z2 = 20 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 = 20 + 12.25 Reorder the terms: 12.25 + 7z + z2 = 20 + 12.25 Combine like terms: 20 + 12.25 = 32.25 12.25 + 7z + z2 = 32.25 Factor a perfect square on the left side: (z + 3.5)(z + 3.5) = 32.25 Calculate the square root of the right side: 5.678908346 Break this problem into two subproblems by setting (z + 3.5) equal to 5.678908346 and -5.678908346.Subproblem 1
z + 3.5 = 5.678908346 Simplifying z + 3.5 = 5.678908346 Reorder the terms: 3.5 + z = 5.678908346 Solving 3.5 + z = 5.678908346 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 = 5.678908346 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 5.678908346 + -3.5 z = 5.678908346 + -3.5 Combine like terms: 5.678908346 + -3.5 = 2.178908346 z = 2.178908346 Simplifying z = 2.178908346Subproblem 2
z + 3.5 = -5.678908346 Simplifying z + 3.5 = -5.678908346 Reorder the terms: 3.5 + z = -5.678908346 Solving 3.5 + z = -5.678908346 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 = -5.678908346 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -5.678908346 + -3.5 z = -5.678908346 + -3.5 Combine like terms: -5.678908346 + -3.5 = -9.178908346 z = -9.178908346 Simplifying z = -9.178908346Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.178908346, -9.178908346}
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