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Simplifying 1000 = 16t2 + 5t Reorder the terms: 1000 = 5t + 16t2 Solving 1000 = 5t + 16t2 Solving for variable 't'. Reorder the terms: 1000 + -5t + -16t2 = 5t + -5t + 16t2 + -16t2 Combine like terms: 5t + -5t = 0 1000 + -5t + -16t2 = 0 + 16t2 + -16t2 1000 + -5t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 1000 + -5t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -62.5 + 0.3125t + t2 = 0 Move the constant term to the right: Add '62.5' to each side of the equation. -62.5 + 0.3125t + 62.5 + t2 = 0 + 62.5 Reorder the terms: -62.5 + 62.5 + 0.3125t + t2 = 0 + 62.5 Combine like terms: -62.5 + 62.5 = 0.0 0.0 + 0.3125t + t2 = 0 + 62.5 0.3125t + t2 = 0 + 62.5 Combine like terms: 0 + 62.5 = 62.5 0.3125t + t2 = 62.5 The t term is 0.3125t. Take half its coefficient (0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. 0.3125t + 0.0244140625 + t2 = 62.5 + 0.0244140625 Reorder the terms: 0.0244140625 + 0.3125t + t2 = 62.5 + 0.0244140625 Combine like terms: 62.5 + 0.0244140625 = 62.5244140625 0.0244140625 + 0.3125t + t2 = 62.5244140625 Factor a perfect square on the left side: (t + 0.15625)(t + 0.15625) = 62.5244140625 Calculate the square root of the right side: 7.907238081 Break this problem into two subproblems by setting (t + 0.15625) equal to 7.907238081 and -7.907238081.Subproblem 1
t + 0.15625 = 7.907238081 Simplifying t + 0.15625 = 7.907238081 Reorder the terms: 0.15625 + t = 7.907238081 Solving 0.15625 + t = 7.907238081 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + t = 7.907238081 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + t = 7.907238081 + -0.15625 t = 7.907238081 + -0.15625 Combine like terms: 7.907238081 + -0.15625 = 7.750988081 t = 7.750988081 Simplifying t = 7.750988081Subproblem 2
t + 0.15625 = -7.907238081 Simplifying t + 0.15625 = -7.907238081 Reorder the terms: 0.15625 + t = -7.907238081 Solving 0.15625 + t = -7.907238081 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + t = -7.907238081 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + t = -7.907238081 + -0.15625 t = -7.907238081 + -0.15625 Combine like terms: -7.907238081 + -0.15625 = -8.063488081 t = -8.063488081 Simplifying t = -8.063488081Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.750988081, -8.063488081}
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