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Simplifying -10t5 + 15t4 + 9t3 = 0 Reorder the terms: 9t3 + 15t4 + -10t5 = 0 Solving 9t3 + 15t4 + -10t5 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't3'. t3(9 + 15t + -10t2) = 0Subproblem 1
Set the factor 't3' equal to zero and attempt to solve: Simplifying t3 = 0 Solving t3 = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t3 = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(9 + 15t + -10t2)' equal to zero and attempt to solve: Simplifying 9 + 15t + -10t2 = 0 Solving 9 + 15t + -10t2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -0.9 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.9' to each side of the equation. -0.9 + -1.5t + 0.9 + t2 = 0 + 0.9 Reorder the terms: -0.9 + 0.9 + -1.5t + t2 = 0 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + -1.5t + t2 = 0 + 0.9 -1.5t + t2 = 0 + 0.9 Combine like terms: 0 + 0.9 = 0.9 -1.5t + t2 = 0.9 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.9 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.9 + 0.5625 Combine like terms: 0.9 + 0.5625 = 1.4625 0.5625 + -1.5t + t2 = 1.4625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 1.4625 Calculate the square root of the right side: 1.209338662 Break this problem into two subproblems by setting (t + -0.75) equal to 1.209338662 and -1.209338662.Subproblem 1
t + -0.75 = 1.209338662 Simplifying t + -0.75 = 1.209338662 Reorder the terms: -0.75 + t = 1.209338662 Solving -0.75 + t = 1.209338662 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 1.209338662 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.209338662 + 0.75 t = 1.209338662 + 0.75 Combine like terms: 1.209338662 + 0.75 = 1.959338662 t = 1.959338662 Simplifying t = 1.959338662Subproblem 2
t + -0.75 = -1.209338662 Simplifying t + -0.75 = -1.209338662 Reorder the terms: -0.75 + t = -1.209338662 Solving -0.75 + t = -1.209338662 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -1.209338662 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.209338662 + 0.75 t = -1.209338662 + 0.75 Combine like terms: -1.209338662 + 0.75 = -0.459338662 t = -0.459338662 Simplifying t = -0.459338662Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.959338662, -0.459338662}Solution
t = {1.959338662, -0.459338662}
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