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Simplifying 90 = -16t2 + 80t + 50 Reorder the terms: 90 = 50 + 80t + -16t2 Solving 90 = 50 + 80t + -16t2 Solving for variable 't'. Combine like terms: 90 + -50 = 40 40 + -80t + 16t2 = 50 + 80t + -16t2 + -50 + -80t + 16t2 Reorder the terms: 40 + -80t + 16t2 = 50 + -50 + 80t + -80t + -16t2 + 16t2 Combine like terms: 50 + -50 = 0 40 + -80t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 40 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 40 + -80t + 16t2 = 0 + -16t2 + 16t2 40 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 40 + -80t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(5 + -10t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(5 + -10t + 2t2)' equal to zero and attempt to solve: Simplifying 5 + -10t + 2t2 = 0 Solving 5 + -10t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + -5t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -5t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -5t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -5t + t2 = 0 + -2.5 -5t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -5t + t2 = -2.5 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = -2.5 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -2.5 + 6.25 Combine like terms: -2.5 + 6.25 = 3.75 6.25 + -5t + t2 = 3.75 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (t + -2.5) equal to 1.936491673 and -1.936491673.Subproblem 1
t + -2.5 = 1.936491673 Simplifying t + -2.5 = 1.936491673 Reorder the terms: -2.5 + t = 1.936491673 Solving -2.5 + t = 1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 1.936491673 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 1.936491673 + 2.5 t = 1.936491673 + 2.5 Combine like terms: 1.936491673 + 2.5 = 4.436491673 t = 4.436491673 Simplifying t = 4.436491673Subproblem 2
t + -2.5 = -1.936491673 Simplifying t + -2.5 = -1.936491673 Reorder the terms: -2.5 + t = -1.936491673 Solving -2.5 + t = -1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -1.936491673 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -1.936491673 + 2.5 t = -1.936491673 + 2.5 Combine like terms: -1.936491673 + 2.5 = 0.563508327 t = 0.563508327 Simplifying t = 0.563508327Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.436491673, 0.563508327}Solution
t = {4.436491673, 0.563508327}
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