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Simplifying 4z2 + -4z + -3 = -6z Reorder the terms: -3 + -4z + 4z2 = -6z Solving -3 + -4z + 4z2 = -6z Solving for variable 'z'. Reorder the terms: -3 + -4z + 6z + 4z2 = -6z + 6z Combine like terms: -4z + 6z = 2z -3 + 2z + 4z2 = -6z + 6z Combine like terms: -6z + 6z = 0 -3 + 2z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 0.5z + z2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.5z + 0.75 + z2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.5z + z2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.5z + z2 = 0 + 0.75 0.5z + z2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.5z + z2 = 0.75 The z term is 0.5z. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5z + 0.0625 + z2 = 0.75 + 0.0625 Reorder the terms: 0.0625 + 0.5z + z2 = 0.75 + 0.0625 Combine like terms: 0.75 + 0.0625 = 0.8125 0.0625 + 0.5z + z2 = 0.8125 Factor a perfect square on the left side: (z + 0.25)(z + 0.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (z + 0.25) equal to 0.901387819 and -0.901387819.Subproblem 1
z + 0.25 = 0.901387819 Simplifying z + 0.25 = 0.901387819 Reorder the terms: 0.25 + z = 0.901387819 Solving 0.25 + z = 0.901387819 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = 0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = 0.901387819 + -0.25 z = 0.901387819 + -0.25 Combine like terms: 0.901387819 + -0.25 = 0.651387819 z = 0.651387819 Simplifying z = 0.651387819Subproblem 2
z + 0.25 = -0.901387819 Simplifying z + 0.25 = -0.901387819 Reorder the terms: 0.25 + z = -0.901387819 Solving 0.25 + z = -0.901387819 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = -0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = -0.901387819 + -0.25 z = -0.901387819 + -0.25 Combine like terms: -0.901387819 + -0.25 = -1.151387819 z = -1.151387819 Simplifying z = -1.151387819Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.651387819, -1.151387819}
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