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Simplifying 5 = z(3z + -1) Reorder the terms: 5 = z(-1 + 3z) 5 = (-1 * z + 3z * z) 5 = (-1z + 3z2) Solving 5 = -1z + 3z2 Solving for variable 'z'. Reorder the terms: 5 + z + -3z2 = -1z + z + 3z2 + -3z2 Combine like terms: -1z + z = 0 5 + z + -3z2 = 0 + 3z2 + -3z2 5 + z + -3z2 = 3z2 + -3z2 Combine like terms: 3z2 + -3z2 = 0 5 + z + -3z2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1.666666667 + -0.3333333333z + z2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + -0.3333333333z + 1.666666667 + z2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + -0.3333333333z + z2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + -0.3333333333z + z2 = 0 + 1.666666667 -0.3333333333z + z2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 -0.3333333333z + z2 = 1.666666667 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.3333333333z + 0.25 + z2 = 1.666666667 + 0.25 Reorder the terms: 0.25 + -0.3333333333z + z2 = 1.666666667 + 0.25 Combine like terms: 1.666666667 + 0.25 = 1.916666667 0.25 + -0.3333333333z + z2 = 1.916666667 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 1.916666667 Calculate the square root of the right side: 1.384437311 Break this problem into two subproblems by setting (z + 0.5) equal to 1.384437311 and -1.384437311.Subproblem 1
z + 0.5 = 1.384437311 Simplifying z + 0.5 = 1.384437311 Reorder the terms: 0.5 + z = 1.384437311 Solving 0.5 + z = 1.384437311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.384437311 + -0.5 z = 1.384437311 + -0.5 Combine like terms: 1.384437311 + -0.5 = 0.884437311 z = 0.884437311 Simplifying z = 0.884437311Subproblem 2
z + 0.5 = -1.384437311 Simplifying z + 0.5 = -1.384437311 Reorder the terms: 0.5 + z = -1.384437311 Solving 0.5 + z = -1.384437311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.384437311 + -0.5 z = -1.384437311 + -0.5 Combine like terms: -1.384437311 + -0.5 = -1.884437311 z = -1.884437311 Simplifying z = -1.884437311Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.884437311, -1.884437311}
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