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Simplifying 3z2 + 15z + 10 = 0 Reorder the terms: 10 + 15z + 3z2 = 0 Solving 10 + 15z + 3z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + 5z + z2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + 5z + -3.333333333 + z2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + 5z + z2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + 5z + z2 = 0 + -3.333333333 5z + z2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 5z + z2 = -3.333333333 The z term is 5z. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5z + 6.25 + z2 = -3.333333333 + 6.25 Reorder the terms: 6.25 + 5z + z2 = -3.333333333 + 6.25 Combine like terms: -3.333333333 + 6.25 = 2.916666667 6.25 + 5z + z2 = 2.916666667 Factor a perfect square on the left side: (z + 2.5)(z + 2.5) = 2.916666667 Calculate the square root of the right side: 1.707825128 Break this problem into two subproblems by setting (z + 2.5) equal to 1.707825128 and -1.707825128.Subproblem 1
z + 2.5 = 1.707825128 Simplifying z + 2.5 = 1.707825128 Reorder the terms: 2.5 + z = 1.707825128 Solving 2.5 + z = 1.707825128 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 1.707825128 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 1.707825128 + -2.5 z = 1.707825128 + -2.5 Combine like terms: 1.707825128 + -2.5 = -0.792174872 z = -0.792174872 Simplifying z = -0.792174872Subproblem 2
z + 2.5 = -1.707825128 Simplifying z + 2.5 = -1.707825128 Reorder the terms: 2.5 + z = -1.707825128 Solving 2.5 + z = -1.707825128 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -1.707825128 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -1.707825128 + -2.5 z = -1.707825128 + -2.5 Combine like terms: -1.707825128 + -2.5 = -4.207825128 z = -4.207825128 Simplifying z = -4.207825128Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.792174872, -4.207825128}
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