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Simplifying 2z2 + 17z + 10 = 0 Reorder the terms: 10 + 17z + 2z2 = 0 Solving 10 + 17z + 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'. 5 + 8.5z + z2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + 8.5z + -5 + z2 = 0 + -5 Reorder the terms: 5 + -5 + 8.5z + z2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 8.5z + z2 = 0 + -5 8.5z + z2 = 0 + -5 Combine like terms: 0 + -5 = -5 8.5z + z2 = -5 The z term is 8.5z. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5z + 18.0625 + z2 = -5 + 18.0625 Reorder the terms: 18.0625 + 8.5z + z2 = -5 + 18.0625 Combine like terms: -5 + 18.0625 = 13.0625 18.0625 + 8.5z + z2 = 13.0625 Factor a perfect square on the left side: (z + 4.25)(z + 4.25) = 13.0625 Calculate the square root of the right side: 3.614208074 Break this problem into two subproblems by setting (z + 4.25) equal to 3.614208074 and -3.614208074.Subproblem 1
z + 4.25 = 3.614208074 Simplifying z + 4.25 = 3.614208074 Reorder the terms: 4.25 + z = 3.614208074 Solving 4.25 + z = 3.614208074 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + z = 3.614208074 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + z = 3.614208074 + -4.25 z = 3.614208074 + -4.25 Combine like terms: 3.614208074 + -4.25 = -0.635791926 z = -0.635791926 Simplifying z = -0.635791926Subproblem 2
z + 4.25 = -3.614208074 Simplifying z + 4.25 = -3.614208074 Reorder the terms: 4.25 + z = -3.614208074 Solving 4.25 + z = -3.614208074 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + z = -3.614208074 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + z = -3.614208074 + -4.25 z = -3.614208074 + -4.25 Combine like terms: -3.614208074 + -4.25 = -7.864208074 z = -7.864208074 Simplifying z = -7.864208074Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.635791926, -7.864208074}
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