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Simplifying 0 = 15t2 + -15t + -135 Reorder the terms: 0 = -135 + -15t + 15t2 Solving 0 = -135 + -15t + 15t2 Solving for variable 't'. Combine like terms: 0 + 135 = 135 135 + 15t + -15t2 = -135 + -15t + 15t2 + 135 + 15t + -15t2 Reorder the terms: 135 + 15t + -15t2 = -135 + 135 + -15t + 15t + 15t2 + -15t2 Combine like terms: -135 + 135 = 0 135 + 15t + -15t2 = 0 + -15t + 15t + 15t2 + -15t2 135 + 15t + -15t2 = -15t + 15t + 15t2 + -15t2 Combine like terms: -15t + 15t = 0 135 + 15t + -15t2 = 0 + 15t2 + -15t2 135 + 15t + -15t2 = 15t2 + -15t2 Combine like terms: 15t2 + -15t2 = 0 135 + 15t + -15t2 = 0 Factor out the Greatest Common Factor (GCF), '15'. 15(9 + t + -1t2) = 0 Ignore the factor 15.Subproblem 1
Set the factor '(9 + t + -1t2)' equal to zero and attempt to solve: Simplifying 9 + t + -1t2 = 0 Solving 9 + t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -9 + -1t + t2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -1t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + -1t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -1t + t2 = 0 + 9 -1t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 -1t + t2 = 9 The t term is t. 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. -1t + 0.25 + t2 = 9 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 9 + 0.25 Combine like terms: 9 + 0.25 = 9.25 0.25 + -1t + t2 = 9.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (t + 0.5) equal to 3.041381265 and -3.041381265.Subproblem 1
t + 0.5 = 3.041381265 Simplifying t + 0.5 = 3.041381265 Reorder the terms: 0.5 + t = 3.041381265 Solving 0.5 + t = 3.041381265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 3.041381265 + -0.5 t = 3.041381265 + -0.5 Combine like terms: 3.041381265 + -0.5 = 2.541381265 t = 2.541381265 Simplifying t = 2.541381265Subproblem 2
t + 0.5 = -3.041381265 Simplifying t + 0.5 = -3.041381265 Reorder the terms: 0.5 + t = -3.041381265 Solving 0.5 + t = -3.041381265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -3.041381265 + -0.5 t = -3.041381265 + -0.5 Combine like terms: -3.041381265 + -0.5 = -3.541381265 t = -3.541381265 Simplifying t = -3.541381265Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.541381265, -3.541381265}Solution
t = {2.541381265, -3.541381265}
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