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Simplifying 3z + 51 = z2 + -5z + -49 Reorder the terms: 51 + 3z = z2 + -5z + -49 Reorder the terms: 51 + 3z = -49 + -5z + z2 Solving 51 + 3z = -49 + -5z + z2 Solving for variable 'z'. Reorder the terms: 51 + 49 + 3z + 5z + -1z2 = -49 + -5z + z2 + 49 + 5z + -1z2 Combine like terms: 51 + 49 = 100 100 + 3z + 5z + -1z2 = -49 + -5z + z2 + 49 + 5z + -1z2 Combine like terms: 3z + 5z = 8z 100 + 8z + -1z2 = -49 + -5z + z2 + 49 + 5z + -1z2 Reorder the terms: 100 + 8z + -1z2 = -49 + 49 + -5z + 5z + z2 + -1z2 Combine like terms: -49 + 49 = 0 100 + 8z + -1z2 = 0 + -5z + 5z + z2 + -1z2 100 + 8z + -1z2 = -5z + 5z + z2 + -1z2 Combine like terms: -5z + 5z = 0 100 + 8z + -1z2 = 0 + z2 + -1z2 100 + 8z + -1z2 = z2 + -1z2 Combine like terms: z2 + -1z2 = 0 100 + 8z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -100 + -8z + z2 = 0 Move the constant term to the right: Add '100' to each side of the equation. -100 + -8z + 100 + z2 = 0 + 100 Reorder the terms: -100 + 100 + -8z + z2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + -8z + z2 = 0 + 100 -8z + z2 = 0 + 100 Combine like terms: 0 + 100 = 100 -8z + z2 = 100 The z term is -8z. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8z + 16 + z2 = 100 + 16 Reorder the terms: 16 + -8z + z2 = 100 + 16 Combine like terms: 100 + 16 = 116 16 + -8z + z2 = 116 Factor a perfect square on the left side: (z + -4)(z + -4) = 116 Calculate the square root of the right side: 10.770329614 Break this problem into two subproblems by setting (z + -4) equal to 10.770329614 and -10.770329614.Subproblem 1
z + -4 = 10.770329614 Simplifying z + -4 = 10.770329614 Reorder the terms: -4 + z = 10.770329614 Solving -4 + z = 10.770329614 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 10.770329614 + 4 Combine like terms: -4 + 4 = 0 0 + z = 10.770329614 + 4 z = 10.770329614 + 4 Combine like terms: 10.770329614 + 4 = 14.770329614 z = 14.770329614 Simplifying z = 14.770329614Subproblem 2
z + -4 = -10.770329614 Simplifying z + -4 = -10.770329614 Reorder the terms: -4 + z = -10.770329614 Solving -4 + z = -10.770329614 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -10.770329614 + 4 Combine like terms: -4 + 4 = 0 0 + z = -10.770329614 + 4 z = -10.770329614 + 4 Combine like terms: -10.770329614 + 4 = -6.770329614 z = -6.770329614 Simplifying z = -6.770329614Solution
The solution to the problem is based on the solutions from the subproblems. z = {14.770329614, -6.770329614}
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