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Simplifying 24z2 + 46z + -55 = 10 Reorder the terms: -55 + 46z + 24z2 = 10 Solving -55 + 46z + 24z2 = 10 Solving for variable 'z'. Reorder the terms: -55 + -10 + 46z + 24z2 = 10 + -10 Combine like terms: -55 + -10 = -65 -65 + 46z + 24z2 = 10 + -10 Combine like terms: 10 + -10 = 0 -65 + 46z + 24z2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -2.708333333 + 1.916666667z + z2 = 0 Move the constant term to the right: Add '2.708333333' to each side of the equation. -2.708333333 + 1.916666667z + 2.708333333 + z2 = 0 + 2.708333333 Reorder the terms: -2.708333333 + 2.708333333 + 1.916666667z + z2 = 0 + 2.708333333 Combine like terms: -2.708333333 + 2.708333333 = 0.000000000 0.000000000 + 1.916666667z + z2 = 0 + 2.708333333 1.916666667z + z2 = 0 + 2.708333333 Combine like terms: 0 + 2.708333333 = 2.708333333 1.916666667z + z2 = 2.708333333 The z term is 1.916666667z. Take half its coefficient (0.9583333335). Square it (0.9184027781) and add it to both sides. Add '0.9184027781' to each side of the equation. 1.916666667z + 0.9184027781 + z2 = 2.708333333 + 0.9184027781 Reorder the terms: 0.9184027781 + 1.916666667z + z2 = 2.708333333 + 0.9184027781 Combine like terms: 2.708333333 + 0.9184027781 = 3.6267361111 0.9184027781 + 1.916666667z + z2 = 3.6267361111 Factor a perfect square on the left side: (z + 0.9583333335)(z + 0.9583333335) = 3.6267361111 Calculate the square root of the right side: 1.904399147 Break this problem into two subproblems by setting (z + 0.9583333335) equal to 1.904399147 and -1.904399147.Subproblem 1
z + 0.9583333335 = 1.904399147 Simplifying z + 0.9583333335 = 1.904399147 Reorder the terms: 0.9583333335 + z = 1.904399147 Solving 0.9583333335 + z = 1.904399147 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.9583333335' to each side of the equation. 0.9583333335 + -0.9583333335 + z = 1.904399147 + -0.9583333335 Combine like terms: 0.9583333335 + -0.9583333335 = 0.0000000000 0.0000000000 + z = 1.904399147 + -0.9583333335 z = 1.904399147 + -0.9583333335 Combine like terms: 1.904399147 + -0.9583333335 = 0.9460658135 z = 0.9460658135 Simplifying z = 0.9460658135Subproblem 2
z + 0.9583333335 = -1.904399147 Simplifying z + 0.9583333335 = -1.904399147 Reorder the terms: 0.9583333335 + z = -1.904399147 Solving 0.9583333335 + z = -1.904399147 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.9583333335' to each side of the equation. 0.9583333335 + -0.9583333335 + z = -1.904399147 + -0.9583333335 Combine like terms: 0.9583333335 + -0.9583333335 = 0.0000000000 0.0000000000 + z = -1.904399147 + -0.9583333335 z = -1.904399147 + -0.9583333335 Combine like terms: -1.904399147 + -0.9583333335 = -2.8627324805 z = -2.8627324805 Simplifying z = -2.8627324805Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.9460658135, -2.8627324805}
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