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Simplifying 4z2 + -7 = -6z Reorder the terms: -7 + 4z2 = -6z Solving -7 + 4z2 = -6z Solving for variable 'z'. Reorder the terms: -7 + 6z + 4z2 = -6z + 6z Combine like terms: -6z + 6z = 0 -7 + 6z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + 1.5z + z2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 1.5z + 1.75 + z2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 1.5z + z2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 1.5z + z2 = 0 + 1.75 1.5z + z2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 1.5z + z2 = 1.75 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 = 1.75 + 0.5625 Reorder the terms: 0.5625 + 1.5z + z2 = 1.75 + 0.5625 Combine like terms: 1.75 + 0.5625 = 2.3125 0.5625 + 1.5z + z2 = 2.3125 Factor a perfect square on the left side: (z + 0.75)(z + 0.75) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (z + 0.75) equal to 1.520690633 and -1.520690633.Subproblem 1
z + 0.75 = 1.520690633 Simplifying z + 0.75 = 1.520690633 Reorder the terms: 0.75 + z = 1.520690633 Solving 0.75 + z = 1.520690633 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 = 1.520690633 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.520690633 + -0.75 z = 1.520690633 + -0.75 Combine like terms: 1.520690633 + -0.75 = 0.770690633 z = 0.770690633 Simplifying z = 0.770690633Subproblem 2
z + 0.75 = -1.520690633 Simplifying z + 0.75 = -1.520690633 Reorder the terms: 0.75 + z = -1.520690633 Solving 0.75 + z = -1.520690633 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 = -1.520690633 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.520690633 + -0.75 z = -1.520690633 + -0.75 Combine like terms: -1.520690633 + -0.75 = -2.270690633 z = -2.270690633 Simplifying z = -2.270690633Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.770690633, -2.270690633}
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