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Simplifying 17.5 = -16t2 + 28t + 7 Reorder the terms: 17.5 = 7 + 28t + -16t2 Solving 17.5 = 7 + 28t + -16t2 Solving for variable 't'. Combine like terms: 17.5 + -7 = 10.5 10.5 + -28t + 16t2 = 7 + 28t + -16t2 + -7 + -28t + 16t2 Reorder the terms: 10.5 + -28t + 16t2 = 7 + -7 + 28t + -28t + -16t2 + 16t2 Combine like terms: 7 + -7 = 0 10.5 + -28t + 16t2 = 0 + 28t + -28t + -16t2 + 16t2 10.5 + -28t + 16t2 = 28t + -28t + -16t2 + 16t2 Combine like terms: 28t + -28t = 0 10.5 + -28t + 16t2 = 0 + -16t2 + 16t2 10.5 + -28t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 10.5 + -28t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.65625 + -1.75t + t2 = 0 Move the constant term to the right: Add '-0.65625' to each side of the equation. 0.65625 + -1.75t + -0.65625 + t2 = 0 + -0.65625 Reorder the terms: 0.65625 + -0.65625 + -1.75t + t2 = 0 + -0.65625 Combine like terms: 0.65625 + -0.65625 = 0.00000 0.00000 + -1.75t + t2 = 0 + -0.65625 -1.75t + t2 = 0 + -0.65625 Combine like terms: 0 + -0.65625 = -0.65625 -1.75t + t2 = -0.65625 The t term is -1.75t. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75t + 0.765625 + t2 = -0.65625 + 0.765625 Reorder the terms: 0.765625 + -1.75t + t2 = -0.65625 + 0.765625 Combine like terms: -0.65625 + 0.765625 = 0.109375 0.765625 + -1.75t + t2 = 0.109375 Factor a perfect square on the left side: (t + -0.875)(t + -0.875) = 0.109375 Calculate the square root of the right side: 0.330718914 Break this problem into two subproblems by setting (t + -0.875) equal to 0.330718914 and -0.330718914.Subproblem 1
t + -0.875 = 0.330718914 Simplifying t + -0.875 = 0.330718914 Reorder the terms: -0.875 + t = 0.330718914 Solving -0.875 + t = 0.330718914 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = 0.330718914 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = 0.330718914 + 0.875 t = 0.330718914 + 0.875 Combine like terms: 0.330718914 + 0.875 = 1.205718914 t = 1.205718914 Simplifying t = 1.205718914Subproblem 2
t + -0.875 = -0.330718914 Simplifying t + -0.875 = -0.330718914 Reorder the terms: -0.875 + t = -0.330718914 Solving -0.875 + t = -0.330718914 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = -0.330718914 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = -0.330718914 + 0.875 t = -0.330718914 + 0.875 Combine like terms: -0.330718914 + 0.875 = 0.544281086 t = 0.544281086 Simplifying t = 0.544281086Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.205718914, 0.544281086}
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