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Simplifying 5z2 + 19z = -4 Reorder the terms: 19z + 5z2 = -4 Solving 19z + 5z2 = -4 Solving for variable 'z'. Reorder the terms: 4 + 19z + 5z2 = -4 + 4 Combine like terms: -4 + 4 = 0 4 + 19z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.8 + 3.8z + z2 = 0 Move the constant term to the right: Add '-0.8' to each side of the equation. 0.8 + 3.8z + -0.8 + z2 = 0 + -0.8 Reorder the terms: 0.8 + -0.8 + 3.8z + z2 = 0 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + 3.8z + z2 = 0 + -0.8 3.8z + z2 = 0 + -0.8 Combine like terms: 0 + -0.8 = -0.8 3.8z + z2 = -0.8 The z term is 3.8z. Take half its coefficient (1.9). Square it (3.61) and add it to both sides. Add '3.61' to each side of the equation. 3.8z + 3.61 + z2 = -0.8 + 3.61 Reorder the terms: 3.61 + 3.8z + z2 = -0.8 + 3.61 Combine like terms: -0.8 + 3.61 = 2.81 3.61 + 3.8z + z2 = 2.81 Factor a perfect square on the left side: (z + 1.9)(z + 1.9) = 2.81 Calculate the square root of the right side: 1.676305461 Break this problem into two subproblems by setting (z + 1.9) equal to 1.676305461 and -1.676305461.Subproblem 1
z + 1.9 = 1.676305461 Simplifying z + 1.9 = 1.676305461 Reorder the terms: 1.9 + z = 1.676305461 Solving 1.9 + z = 1.676305461 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + z = 1.676305461 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + z = 1.676305461 + -1.9 z = 1.676305461 + -1.9 Combine like terms: 1.676305461 + -1.9 = -0.223694539 z = -0.223694539 Simplifying z = -0.223694539Subproblem 2
z + 1.9 = -1.676305461 Simplifying z + 1.9 = -1.676305461 Reorder the terms: 1.9 + z = -1.676305461 Solving 1.9 + z = -1.676305461 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + z = -1.676305461 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + z = -1.676305461 + -1.9 z = -1.676305461 + -1.9 Combine like terms: -1.676305461 + -1.9 = -3.576305461 z = -3.576305461 Simplifying z = -3.576305461Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.223694539, -3.576305461}
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