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Simplifying z2 + -24z = 47 Reorder the terms: -24z + z2 = 47 Solving -24z + z2 = 47 Solving for variable 'z'. Reorder the terms: -47 + -24z + z2 = 47 + -47 Combine like terms: 47 + -47 = 0 -47 + -24z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '47' to each side of the equation. -47 + -24z + 47 + z2 = 0 + 47 Reorder the terms: -47 + 47 + -24z + z2 = 0 + 47 Combine like terms: -47 + 47 = 0 0 + -24z + z2 = 0 + 47 -24z + z2 = 0 + 47 Combine like terms: 0 + 47 = 47 -24z + z2 = 47 The z term is -24z. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24z + 144 + z2 = 47 + 144 Reorder the terms: 144 + -24z + z2 = 47 + 144 Combine like terms: 47 + 144 = 191 144 + -24z + z2 = 191 Factor a perfect square on the left side: (z + -12)(z + -12) = 191 Calculate the square root of the right side: 13.820274961 Break this problem into two subproblems by setting (z + -12) equal to 13.820274961 and -13.820274961.Subproblem 1
z + -12 = 13.820274961 Simplifying z + -12 = 13.820274961 Reorder the terms: -12 + z = 13.820274961 Solving -12 + z = 13.820274961 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + z = 13.820274961 + 12 Combine like terms: -12 + 12 = 0 0 + z = 13.820274961 + 12 z = 13.820274961 + 12 Combine like terms: 13.820274961 + 12 = 25.820274961 z = 25.820274961 Simplifying z = 25.820274961Subproblem 2
z + -12 = -13.820274961 Simplifying z + -12 = -13.820274961 Reorder the terms: -12 + z = -13.820274961 Solving -12 + z = -13.820274961 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + z = -13.820274961 + 12 Combine like terms: -12 + 12 = 0 0 + z = -13.820274961 + 12 z = -13.820274961 + 12 Combine like terms: -13.820274961 + 12 = -1.820274961 z = -1.820274961 Simplifying z = -1.820274961Solution
The solution to the problem is based on the solutions from the subproblems. z = {25.820274961, -1.820274961}
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