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Simplifying 2z2 + z = 13 Reorder the terms: z + 2z2 = 13 Solving z + 2z2 = 13 Solving for variable 'z'. Reorder the terms: -13 + z + 2z2 = 13 + -13 Combine like terms: 13 + -13 = 0 -13 + z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + 0.5z + z2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + 0.5z + 6.5 + z2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + 0.5z + z2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + 0.5z + z2 = 0 + 6.5 0.5z + z2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 0.5z + z2 = 6.5 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. 0.5z + 0.25 + z2 = 6.5 + 0.25 Reorder the terms: 0.25 + 0.5z + z2 = 6.5 + 0.25 Combine like terms: 6.5 + 0.25 = 6.75 0.25 + 0.5z + z2 = 6.75 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (z + 0.5) equal to 2.598076211 and -2.598076211.Subproblem 1
z + 0.5 = 2.598076211 Simplifying z + 0.5 = 2.598076211 Reorder the terms: 0.5 + z = 2.598076211 Solving 0.5 + z = 2.598076211 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 = 2.598076211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 2.598076211 + -0.5 z = 2.598076211 + -0.5 Combine like terms: 2.598076211 + -0.5 = 2.098076211 z = 2.098076211 Simplifying z = 2.098076211Subproblem 2
z + 0.5 = -2.598076211 Simplifying z + 0.5 = -2.598076211 Reorder the terms: 0.5 + z = -2.598076211 Solving 0.5 + z = -2.598076211 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 = -2.598076211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -2.598076211 + -0.5 z = -2.598076211 + -0.5 Combine like terms: -2.598076211 + -0.5 = -3.098076211 z = -3.098076211 Simplifying z = -3.098076211Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.098076211, -3.098076211}
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