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Simplifying 15 = 16t2 + 32t + 3 Reorder the terms: 15 = 3 + 32t + 16t2 Solving 15 = 3 + 32t + 16t2 Solving for variable 't'. Combine like terms: 15 + -3 = 12 12 + -32t + -16t2 = 3 + 32t + 16t2 + -3 + -32t + -16t2 Reorder the terms: 12 + -32t + -16t2 = 3 + -3 + 32t + -32t + 16t2 + -16t2 Combine like terms: 3 + -3 = 0 12 + -32t + -16t2 = 0 + 32t + -32t + 16t2 + -16t2 12 + -32t + -16t2 = 32t + -32t + 16t2 + -16t2 Combine like terms: 32t + -32t = 0 12 + -32t + -16t2 = 0 + 16t2 + -16t2 12 + -32t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 12 + -32t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(3 + -8t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(3 + -8t + -4t2)' equal to zero and attempt to solve: Simplifying 3 + -8t + -4t2 = 0 Solving 3 + -8t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + 2t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 2t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 2t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 2t + t2 = 0 + 0.75 2t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 2t + t2 = 0.75 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 0.75 + 1 Reorder the terms: 1 + 2t + t2 = 0.75 + 1 Combine like terms: 0.75 + 1 = 1.75 1 + 2t + t2 = 1.75 Factor a perfect square on the left side: (t + 1)(t + 1) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (t + 1) equal to 1.322875656 and -1.322875656.Subproblem 1
t + 1 = 1.322875656 Simplifying t + 1 = 1.322875656 Reorder the terms: 1 + t = 1.322875656 Solving 1 + t = 1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 1.322875656 + -1 Combine like terms: 1 + -1 = 0 0 + t = 1.322875656 + -1 t = 1.322875656 + -1 Combine like terms: 1.322875656 + -1 = 0.322875656 t = 0.322875656 Simplifying t = 0.322875656Subproblem 2
t + 1 = -1.322875656 Simplifying t + 1 = -1.322875656 Reorder the terms: 1 + t = -1.322875656 Solving 1 + t = -1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -1.322875656 + -1 Combine like terms: 1 + -1 = 0 0 + t = -1.322875656 + -1 t = -1.322875656 + -1 Combine like terms: -1.322875656 + -1 = -2.322875656 t = -2.322875656 Simplifying t = -2.322875656Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.322875656, -2.322875656}Solution
t = {0.322875656, -2.322875656}
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