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Simplifying 10z2 + 23z + 5 = 0 Reorder the terms: 5 + 23z + 10z2 = 0 Solving 5 + 23z + 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'. 0.5 + 2.3z + z2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2.3z + -0.5 + z2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2.3z + z2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2.3z + z2 = 0 + -0.5 2.3z + z2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2.3z + z2 = -0.5 The z term is 2.3z. Take half its coefficient (1.15). Square it (1.3225) and add it to both sides. Add '1.3225' to each side of the equation. 2.3z + 1.3225 + z2 = -0.5 + 1.3225 Reorder the terms: 1.3225 + 2.3z + z2 = -0.5 + 1.3225 Combine like terms: -0.5 + 1.3225 = 0.8225 1.3225 + 2.3z + z2 = 0.8225 Factor a perfect square on the left side: (z + 1.15)(z + 1.15) = 0.8225 Calculate the square root of the right side: 0.906917857 Break this problem into two subproblems by setting (z + 1.15) equal to 0.906917857 and -0.906917857.Subproblem 1
z + 1.15 = 0.906917857 Simplifying z + 1.15 = 0.906917857 Reorder the terms: 1.15 + z = 0.906917857 Solving 1.15 + z = 0.906917857 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.15' to each side of the equation. 1.15 + -1.15 + z = 0.906917857 + -1.15 Combine like terms: 1.15 + -1.15 = 0.00 0.00 + z = 0.906917857 + -1.15 z = 0.906917857 + -1.15 Combine like terms: 0.906917857 + -1.15 = -0.243082143 z = -0.243082143 Simplifying z = -0.243082143Subproblem 2
z + 1.15 = -0.906917857 Simplifying z + 1.15 = -0.906917857 Reorder the terms: 1.15 + z = -0.906917857 Solving 1.15 + z = -0.906917857 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.15' to each side of the equation. 1.15 + -1.15 + z = -0.906917857 + -1.15 Combine like terms: 1.15 + -1.15 = 0.00 0.00 + z = -0.906917857 + -1.15 z = -0.906917857 + -1.15 Combine like terms: -0.906917857 + -1.15 = -2.056917857 z = -2.056917857 Simplifying z = -2.056917857Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.243082143, -2.056917857}
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