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Simplifying t = -16t2 + 10t + 20 Reorder the terms: t = 20 + 10t + -16t2 Solving t = 20 + 10t + -16t2 Solving for variable 't'. Reorder the terms: -20 + t + -10t + 16t2 = 20 + 10t + -16t2 + -20 + -10t + 16t2 Combine like terms: t + -10t = -9t -20 + -9t + 16t2 = 20 + 10t + -16t2 + -20 + -10t + 16t2 Reorder the terms: -20 + -9t + 16t2 = 20 + -20 + 10t + -10t + -16t2 + 16t2 Combine like terms: 20 + -20 = 0 -20 + -9t + 16t2 = 0 + 10t + -10t + -16t2 + 16t2 -20 + -9t + 16t2 = 10t + -10t + -16t2 + 16t2 Combine like terms: 10t + -10t = 0 -20 + -9t + 16t2 = 0 + -16t2 + 16t2 -20 + -9t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -20 + -9t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.25 + -0.5625t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -0.5625t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -0.5625t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -0.5625t + t2 = 0 + 1.25 -0.5625t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -0.5625t + t2 = 1.25 The t term is -0.5625t. Take half its coefficient (-0.28125). Square it (0.0791015625) and add it to both sides. Add '0.0791015625' to each side of the equation. -0.5625t + 0.0791015625 + t2 = 1.25 + 0.0791015625 Reorder the terms: 0.0791015625 + -0.5625t + t2 = 1.25 + 0.0791015625 Combine like terms: 1.25 + 0.0791015625 = 1.3291015625 0.0791015625 + -0.5625t + t2 = 1.3291015625 Factor a perfect square on the left side: (t + -0.28125)(t + -0.28125) = 1.3291015625 Calculate the square root of the right side: 1.152866672 Break this problem into two subproblems by setting (t + -0.28125) equal to 1.152866672 and -1.152866672.Subproblem 1
t + -0.28125 = 1.152866672 Simplifying t + -0.28125 = 1.152866672 Reorder the terms: -0.28125 + t = 1.152866672 Solving -0.28125 + t = 1.152866672 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.28125' to each side of the equation. -0.28125 + 0.28125 + t = 1.152866672 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + t = 1.152866672 + 0.28125 t = 1.152866672 + 0.28125 Combine like terms: 1.152866672 + 0.28125 = 1.434116672 t = 1.434116672 Simplifying t = 1.434116672Subproblem 2
t + -0.28125 = -1.152866672 Simplifying t + -0.28125 = -1.152866672 Reorder the terms: -0.28125 + t = -1.152866672 Solving -0.28125 + t = -1.152866672 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.28125' to each side of the equation. -0.28125 + 0.28125 + t = -1.152866672 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + t = -1.152866672 + 0.28125 t = -1.152866672 + 0.28125 Combine like terms: -1.152866672 + 0.28125 = -0.871616672 t = -0.871616672 Simplifying t = -0.871616672Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.434116672, -0.871616672}
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