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Simplifying z2 + -8z = 47 Reorder the terms: -8z + z2 = 47 Solving -8z + z2 = 47 Solving for variable 'z'. Reorder the terms: -47 + -8z + z2 = 47 + -47 Combine like terms: 47 + -47 = 0 -47 + -8z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '47' to each side of the equation. -47 + -8z + 47 + z2 = 0 + 47 Reorder the terms: -47 + 47 + -8z + z2 = 0 + 47 Combine like terms: -47 + 47 = 0 0 + -8z + z2 = 0 + 47 -8z + z2 = 0 + 47 Combine like terms: 0 + 47 = 47 -8z + z2 = 47 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 = 47 + 16 Reorder the terms: 16 + -8z + z2 = 47 + 16 Combine like terms: 47 + 16 = 63 16 + -8z + z2 = 63 Factor a perfect square on the left side: (z + -4)(z + -4) = 63 Calculate the square root of the right side: 7.937253933 Break this problem into two subproblems by setting (z + -4) equal to 7.937253933 and -7.937253933.Subproblem 1
z + -4 = 7.937253933 Simplifying z + -4 = 7.937253933 Reorder the terms: -4 + z = 7.937253933 Solving -4 + z = 7.937253933 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 = 7.937253933 + 4 Combine like terms: -4 + 4 = 0 0 + z = 7.937253933 + 4 z = 7.937253933 + 4 Combine like terms: 7.937253933 + 4 = 11.937253933 z = 11.937253933 Simplifying z = 11.937253933Subproblem 2
z + -4 = -7.937253933 Simplifying z + -4 = -7.937253933 Reorder the terms: -4 + z = -7.937253933 Solving -4 + z = -7.937253933 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 = -7.937253933 + 4 Combine like terms: -4 + 4 = 0 0 + z = -7.937253933 + 4 z = -7.937253933 + 4 Combine like terms: -7.937253933 + 4 = -3.937253933 z = -3.937253933 Simplifying z = -3.937253933Solution
The solution to the problem is based on the solutions from the subproblems. z = {11.937253933, -3.937253933}
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