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Simplifying 24z2 + 46z + -15 + -10 = 0 Reorder the terms: -15 + -10 + 46z + 24z2 = 0 Combine like terms: -15 + -10 = -25 -25 + 46z + 24z2 = 0 Solving -25 + 46z + 24z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -1.041666667 + 1.916666667z + z2 = 0 Move the constant term to the right: Add '1.041666667' to each side of the equation. -1.041666667 + 1.916666667z + 1.041666667 + z2 = 0 + 1.041666667 Reorder the terms: -1.041666667 + 1.041666667 + 1.916666667z + z2 = 0 + 1.041666667 Combine like terms: -1.041666667 + 1.041666667 = 0.000000000 0.000000000 + 1.916666667z + z2 = 0 + 1.041666667 1.916666667z + z2 = 0 + 1.041666667 Combine like terms: 0 + 1.041666667 = 1.041666667 1.916666667z + z2 = 1.041666667 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 = 1.041666667 + 0.9184027781 Reorder the terms: 0.9184027781 + 1.916666667z + z2 = 1.041666667 + 0.9184027781 Combine like terms: 1.041666667 + 0.9184027781 = 1.9600694451 0.9184027781 + 1.916666667z + z2 = 1.9600694451 Factor a perfect square on the left side: (z + 0.9583333335)(z + 0.9583333335) = 1.9600694451 Calculate the square root of the right side: 1.400024802 Break this problem into two subproblems by setting (z + 0.9583333335) equal to 1.400024802 and -1.400024802.Subproblem 1
z + 0.9583333335 = 1.400024802 Simplifying z + 0.9583333335 = 1.400024802 Reorder the terms: 0.9583333335 + z = 1.400024802 Solving 0.9583333335 + z = 1.400024802 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 = 1.400024802 + -0.9583333335 Combine like terms: 0.9583333335 + -0.9583333335 = 0.0000000000 0.0000000000 + z = 1.400024802 + -0.9583333335 z = 1.400024802 + -0.9583333335 Combine like terms: 1.400024802 + -0.9583333335 = 0.4416914685 z = 0.4416914685 Simplifying z = 0.4416914685Subproblem 2
z + 0.9583333335 = -1.400024802 Simplifying z + 0.9583333335 = -1.400024802 Reorder the terms: 0.9583333335 + z = -1.400024802 Solving 0.9583333335 + z = -1.400024802 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 = -1.400024802 + -0.9583333335 Combine like terms: 0.9583333335 + -0.9583333335 = 0.0000000000 0.0000000000 + z = -1.400024802 + -0.9583333335 z = -1.400024802 + -0.9583333335 Combine like terms: -1.400024802 + -0.9583333335 = -2.3583581355 z = -2.3583581355 Simplifying z = -2.3583581355Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.4416914685, -2.3583581355}
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