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Simplifying -2z2 + -2z + 9 = 0 Reorder the terms: 9 + -2z + -2z2 = 0 Solving 9 + -2z + -2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -4.5 + z + z2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + z + 4.5 + z2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + z + z2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z + z2 = 0 + 4.5 z + z2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 z + z2 = 4.5 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + z2 = 4.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + z2 = 4.5 + 0.25 Combine like terms: 4.5 + 0.25 = 4.75 1.25 + z2 = 4.75 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (z + 0.5) equal to 2.179449472 and -2.179449472.Subproblem 1
z + 0.5 = 2.179449472 Simplifying z + 0.5 = 2.179449472 Reorder the terms: 0.5 + z = 2.179449472 Solving 0.5 + z = 2.179449472 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 2.179449472 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 2.179449472 + -0.5 z = 2.179449472 + -0.5 Combine like terms: 2.179449472 + -0.5 = 1.679449472 z = 1.679449472 Simplifying z = 1.679449472Subproblem 2
z + 0.5 = -2.179449472 Simplifying z + 0.5 = -2.179449472 Reorder the terms: 0.5 + z = -2.179449472 Solving 0.5 + z = -2.179449472 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -2.179449472 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -2.179449472 + -0.5 z = -2.179449472 + -0.5 Combine like terms: -2.179449472 + -0.5 = -2.679449472 z = -2.679449472 Simplifying z = -2.679449472Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.679449472, -2.679449472}
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