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Simplifying 4z2 + 6z + -3 = 0 Reorder the terms: -3 + 6z + 4z2 = 0 Solving -3 + 6z + 4z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 1.5z + z2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.5z + 0.75 + z2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.5z + z2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.5z + z2 = 0 + 0.75 1.5z + z2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.5z + z2 = 0.75 The z term is 1.5z. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5z + 0.5625 + z2 = 0.75 + 0.5625 Reorder the terms: 0.5625 + 1.5z + z2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + 1.5z + z2 = 1.3125 Factor a perfect square on the left side: (z + 0.75)(z + 0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (z + 0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
z + 0.75 = 1.145643924 Simplifying z + 0.75 = 1.145643924 Reorder the terms: 0.75 + z = 1.145643924 Solving 0.75 + z = 1.145643924 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = 1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.145643924 + -0.75 z = 1.145643924 + -0.75 Combine like terms: 1.145643924 + -0.75 = 0.395643924 z = 0.395643924 Simplifying z = 0.395643924Subproblem 2
z + 0.75 = -1.145643924 Simplifying z + 0.75 = -1.145643924 Reorder the terms: 0.75 + z = -1.145643924 Solving 0.75 + z = -1.145643924 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = -1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.145643924 + -0.75 z = -1.145643924 + -0.75 Combine like terms: -1.145643924 + -0.75 = -1.895643924 z = -1.895643924 Simplifying z = -1.895643924Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.395643924, -1.895643924}
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