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Simplifying -12s2 + 4s + 128000 = 0 Reorder the terms: 128000 + 4s + -12s2 = 0 Solving 128000 + 4s + -12s2 = 0 Solving for variable 's'. Factor out the Greatest Common Factor (GCF), '4'. 4(32000 + s + -3s2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(32000 + s + -3s2)' equal to zero and attempt to solve: Simplifying 32000 + s + -3s2 = 0 Solving 32000 + s + -3s2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -10666.66667 + -0.3333333333s + s2 = 0 Move the constant term to the right: Add '10666.66667' to each side of the equation. -10666.66667 + -0.3333333333s + 10666.66667 + s2 = 0 + 10666.66667 Reorder the terms: -10666.66667 + 10666.66667 + -0.3333333333s + s2 = 0 + 10666.66667 Combine like terms: -10666.66667 + 10666.66667 = 0.00000 0.00000 + -0.3333333333s + s2 = 0 + 10666.66667 -0.3333333333s + s2 = 0 + 10666.66667 Combine like terms: 0 + 10666.66667 = 10666.66667 -0.3333333333s + s2 = 10666.66667 The s term is s. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.3333333333s + 0.25 + s2 = 10666.66667 + 0.25 Reorder the terms: 0.25 + -0.3333333333s + s2 = 10666.66667 + 0.25 Combine like terms: 10666.66667 + 0.25 = 10666.91667 0.25 + -0.3333333333s + s2 = 10666.91667 Factor a perfect square on the left side: (s + 0.5)(s + 0.5) = 10666.91667 Calculate the square root of the right side: 103.280766215 Break this problem into two subproblems by setting (s + 0.5) equal to 103.280766215 and -103.280766215.Subproblem 1
s + 0.5 = 103.280766215 Simplifying s + 0.5 = 103.280766215 Reorder the terms: 0.5 + s = 103.280766215 Solving 0.5 + s = 103.280766215 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + s = 103.280766215 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + s = 103.280766215 + -0.5 s = 103.280766215 + -0.5 Combine like terms: 103.280766215 + -0.5 = 102.780766215 s = 102.780766215 Simplifying s = 102.780766215Subproblem 2
s + 0.5 = -103.280766215 Simplifying s + 0.5 = -103.280766215 Reorder the terms: 0.5 + s = -103.280766215 Solving 0.5 + s = -103.280766215 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + s = -103.280766215 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + s = -103.280766215 + -0.5 s = -103.280766215 + -0.5 Combine like terms: -103.280766215 + -0.5 = -103.780766215 s = -103.780766215 Simplifying s = -103.780766215Solution
The solution to the problem is based on the solutions from the subproblems. s = {102.780766215, -103.780766215}Solution
s = {102.780766215, -103.780766215}
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