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Simplifying 2z * 2z + 3z = 135 Reorder the terms for easier multiplication: 2 * 2z * z + 3z = 135 Multiply 2 * 2 4z * z + 3z = 135 Multiply z * z 4z2 + 3z = 135 Reorder the terms: 3z + 4z2 = 135 Solving 3z + 4z2 = 135 Solving for variable 'z'. Reorder the terms: -135 + 3z + 4z2 = 135 + -135 Combine like terms: 135 + -135 = 0 -135 + 3z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -33.75 + 0.75z + z2 = 0 Move the constant term to the right: Add '33.75' to each side of the equation. -33.75 + 0.75z + 33.75 + z2 = 0 + 33.75 Reorder the terms: -33.75 + 33.75 + 0.75z + z2 = 0 + 33.75 Combine like terms: -33.75 + 33.75 = 0.00 0.00 + 0.75z + z2 = 0 + 33.75 0.75z + z2 = 0 + 33.75 Combine like terms: 0 + 33.75 = 33.75 0.75z + z2 = 33.75 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 = 33.75 + 0.140625 Reorder the terms: 0.140625 + 0.75z + z2 = 33.75 + 0.140625 Combine like terms: 33.75 + 0.140625 = 33.890625 0.140625 + 0.75z + z2 = 33.890625 Factor a perfect square on the left side: (z + 0.375)(z + 0.375) = 33.890625 Calculate the square root of the right side: 5.821565511 Break this problem into two subproblems by setting (z + 0.375) equal to 5.821565511 and -5.821565511.Subproblem 1
z + 0.375 = 5.821565511 Simplifying z + 0.375 = 5.821565511 Reorder the terms: 0.375 + z = 5.821565511 Solving 0.375 + z = 5.821565511 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 = 5.821565511 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = 5.821565511 + -0.375 z = 5.821565511 + -0.375 Combine like terms: 5.821565511 + -0.375 = 5.446565511 z = 5.446565511 Simplifying z = 5.446565511Subproblem 2
z + 0.375 = -5.821565511 Simplifying z + 0.375 = -5.821565511 Reorder the terms: 0.375 + z = -5.821565511 Solving 0.375 + z = -5.821565511 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 = -5.821565511 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = -5.821565511 + -0.375 z = -5.821565511 + -0.375 Combine like terms: -5.821565511 + -0.375 = -6.196565511 z = -6.196565511 Simplifying z = -6.196565511Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.446565511, -6.196565511}
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