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Simplifying 13 = 4z2 + -3z Reorder the terms: 13 = -3z + 4z2 Solving 13 = -3z + 4z2 Solving for variable 'z'. Reorder the terms: 13 + 3z + -4z2 = -3z + 3z + 4z2 + -4z2 Combine like terms: -3z + 3z = 0 13 + 3z + -4z2 = 0 + 4z2 + -4z2 13 + 3z + -4z2 = 4z2 + -4z2 Combine like terms: 4z2 + -4z2 = 0 13 + 3z + -4z2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -3.25 + -0.75z + z2 = 0 Move the constant term to the right: Add '3.25' to each side of the equation. -3.25 + -0.75z + 3.25 + z2 = 0 + 3.25 Reorder the terms: -3.25 + 3.25 + -0.75z + z2 = 0 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + -0.75z + z2 = 0 + 3.25 -0.75z + z2 = 0 + 3.25 Combine like terms: 0 + 3.25 = 3.25 -0.75z + z2 = 3.25 The z term is -0.75z. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75z + 0.140625 + z2 = 3.25 + 0.140625 Reorder the terms: 0.140625 + -0.75z + z2 = 3.25 + 0.140625 Combine like terms: 3.25 + 0.140625 = 3.390625 0.140625 + -0.75z + z2 = 3.390625 Factor a perfect square on the left side: (z + -0.375)(z + -0.375) = 3.390625 Calculate the square root of the right side: 1.841364983 Break this problem into two subproblems by setting (z + -0.375) equal to 1.841364983 and -1.841364983.Subproblem 1
z + -0.375 = 1.841364983 Simplifying z + -0.375 = 1.841364983 Reorder the terms: -0.375 + z = 1.841364983 Solving -0.375 + z = 1.841364983 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + z = 1.841364983 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + z = 1.841364983 + 0.375 z = 1.841364983 + 0.375 Combine like terms: 1.841364983 + 0.375 = 2.216364983 z = 2.216364983 Simplifying z = 2.216364983Subproblem 2
z + -0.375 = -1.841364983 Simplifying z + -0.375 = -1.841364983 Reorder the terms: -0.375 + z = -1.841364983 Solving -0.375 + z = -1.841364983 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + z = -1.841364983 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + z = -1.841364983 + 0.375 z = -1.841364983 + 0.375 Combine like terms: -1.841364983 + 0.375 = -1.466364983 z = -1.466364983 Simplifying z = -1.466364983Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.216364983, -1.466364983}
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