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Simplifying z2 + -5z + 3 = 0 Reorder the terms: 3 + -5z + z2 = 0 Solving 3 + -5z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -5z + -3 + z2 = 0 + -3 Reorder the terms: 3 + -3 + -5z + z2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -5z + z2 = 0 + -3 -5z + z2 = 0 + -3 Combine like terms: 0 + -3 = -3 -5z + z2 = -3 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 + 6.25 Reorder the terms: 6.25 + -5z + z2 = -3 + 6.25 Combine like terms: -3 + 6.25 = 3.25 6.25 + -5z + z2 = 3.25 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (z + -2.5) equal to 1.802775638 and -1.802775638.Subproblem 1
z + -2.5 = 1.802775638 Simplifying z + -2.5 = 1.802775638 Reorder the terms: -2.5 + z = 1.802775638 Solving -2.5 + z = 1.802775638 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 = 1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 1.802775638 + 2.5 z = 1.802775638 + 2.5 Combine like terms: 1.802775638 + 2.5 = 4.302775638 z = 4.302775638 Simplifying z = 4.302775638Subproblem 2
z + -2.5 = -1.802775638 Simplifying z + -2.5 = -1.802775638 Reorder the terms: -2.5 + z = -1.802775638 Solving -2.5 + z = -1.802775638 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 = -1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -1.802775638 + 2.5 z = -1.802775638 + 2.5 Combine like terms: -1.802775638 + 2.5 = 0.697224362 z = 0.697224362 Simplifying z = 0.697224362Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.302775638, 0.697224362}
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