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Simplifying -5t2 + 135t + 10 = 0 Reorder the terms: 10 + 135t + -5t2 = 0 Solving 10 + 135t + -5t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5'. 5(2 + 27t + -1t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(2 + 27t + -1t2)' equal to zero and attempt to solve: Simplifying 2 + 27t + -1t2 = 0 Solving 2 + 27t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + -27t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -27t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + -27t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -27t + t2 = 0 + 2 -27t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 -27t + t2 = 2 The t term is -27t. Take half its coefficient (-13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. -27t + 182.25 + t2 = 2 + 182.25 Reorder the terms: 182.25 + -27t + t2 = 2 + 182.25 Combine like terms: 2 + 182.25 = 184.25 182.25 + -27t + t2 = 184.25 Factor a perfect square on the left side: (t + -13.5)(t + -13.5) = 184.25 Calculate the square root of the right side: 13.57387196 Break this problem into two subproblems by setting (t + -13.5) equal to 13.57387196 and -13.57387196.Subproblem 1
t + -13.5 = 13.57387196 Simplifying t + -13.5 = 13.57387196 Reorder the terms: -13.5 + t = 13.57387196 Solving -13.5 + t = 13.57387196 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + t = 13.57387196 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + t = 13.57387196 + 13.5 t = 13.57387196 + 13.5 Combine like terms: 13.57387196 + 13.5 = 27.07387196 t = 27.07387196 Simplifying t = 27.07387196Subproblem 2
t + -13.5 = -13.57387196 Simplifying t + -13.5 = -13.57387196 Reorder the terms: -13.5 + t = -13.57387196 Solving -13.5 + t = -13.57387196 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + t = -13.57387196 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + t = -13.57387196 + 13.5 t = -13.57387196 + 13.5 Combine like terms: -13.57387196 + 13.5 = -0.07387196 t = -0.07387196 Simplifying t = -0.07387196Solution
The solution to the problem is based on the solutions from the subproblems. t = {27.07387196, -0.07387196}Solution
t = {27.07387196, -0.07387196}
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