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Simplifying -2t2 + 30t + 6.5 = 12 Reorder the terms: 6.5 + 30t + -2t2 = 12 Solving 6.5 + 30t + -2t2 = 12 Solving for variable 't'. Reorder the terms: 6.5 + -12 + 30t + -2t2 = 12 + -12 Combine like terms: 6.5 + -12 = -5.5 -5.5 + 30t + -2t2 = 12 + -12 Combine like terms: 12 + -12 = 0 -5.5 + 30t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 2.75 + -15t + t2 = 0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + -15t + -2.75 + t2 = 0 + -2.75 Reorder the terms: 2.75 + -2.75 + -15t + t2 = 0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + -15t + t2 = 0 + -2.75 -15t + t2 = 0 + -2.75 Combine like terms: 0 + -2.75 = -2.75 -15t + t2 = -2.75 The t term is -15t. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15t + 56.25 + t2 = -2.75 + 56.25 Reorder the terms: 56.25 + -15t + t2 = -2.75 + 56.25 Combine like terms: -2.75 + 56.25 = 53.5 56.25 + -15t + t2 = 53.5 Factor a perfect square on the left side: (t + -7.5)(t + -7.5) = 53.5 Calculate the square root of the right side: 7.314369419 Break this problem into two subproblems by setting (t + -7.5) equal to 7.314369419 and -7.314369419.Subproblem 1
t + -7.5 = 7.314369419 Simplifying t + -7.5 = 7.314369419 Reorder the terms: -7.5 + t = 7.314369419 Solving -7.5 + t = 7.314369419 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = 7.314369419 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 7.314369419 + 7.5 t = 7.314369419 + 7.5 Combine like terms: 7.314369419 + 7.5 = 14.814369419 t = 14.814369419 Simplifying t = 14.814369419Subproblem 2
t + -7.5 = -7.314369419 Simplifying t + -7.5 = -7.314369419 Reorder the terms: -7.5 + t = -7.314369419 Solving -7.5 + t = -7.314369419 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = -7.314369419 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -7.314369419 + 7.5 t = -7.314369419 + 7.5 Combine like terms: -7.314369419 + 7.5 = 0.185630581 t = 0.185630581 Simplifying t = 0.185630581Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.814369419, 0.185630581}
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