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Simplifying 2z2 + 19z + 20 = 0 Reorder the terms: 20 + 19z + 2z2 = 0 Solving 20 + 19z + 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'. 10 + 9.5z + z2 = 0 Move the constant term to the right: Add '-10' to each side of the equation. 10 + 9.5z + -10 + z2 = 0 + -10 Reorder the terms: 10 + -10 + 9.5z + z2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + 9.5z + z2 = 0 + -10 9.5z + z2 = 0 + -10 Combine like terms: 0 + -10 = -10 9.5z + z2 = -10 The z term is 9.5z. Take half its coefficient (4.75). Square it (22.5625) and add it to both sides. Add '22.5625' to each side of the equation. 9.5z + 22.5625 + z2 = -10 + 22.5625 Reorder the terms: 22.5625 + 9.5z + z2 = -10 + 22.5625 Combine like terms: -10 + 22.5625 = 12.5625 22.5625 + 9.5z + z2 = 12.5625 Factor a perfect square on the left side: (z + 4.75)(z + 4.75) = 12.5625 Calculate the square root of the right side: 3.54436172 Break this problem into two subproblems by setting (z + 4.75) equal to 3.54436172 and -3.54436172.Subproblem 1
z + 4.75 = 3.54436172 Simplifying z + 4.75 = 3.54436172 Reorder the terms: 4.75 + z = 3.54436172 Solving 4.75 + z = 3.54436172 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + z = 3.54436172 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + z = 3.54436172 + -4.75 z = 3.54436172 + -4.75 Combine like terms: 3.54436172 + -4.75 = -1.20563828 z = -1.20563828 Simplifying z = -1.20563828Subproblem 2
z + 4.75 = -3.54436172 Simplifying z + 4.75 = -3.54436172 Reorder the terms: 4.75 + z = -3.54436172 Solving 4.75 + z = -3.54436172 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + z = -3.54436172 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + z = -3.54436172 + -4.75 z = -3.54436172 + -4.75 Combine like terms: -3.54436172 + -4.75 = -8.29436172 z = -8.29436172 Simplifying z = -8.29436172Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.20563828, -8.29436172}
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