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Simplifying 123(5) = 16t2 + 123t Multiply 123 * 5 615 = 16t2 + 123t Reorder the terms: 615 = 123t + 16t2 Solving 615 = 123t + 16t2 Solving for variable 't'. Reorder the terms: 615 + -123t + -16t2 = 123t + -123t + 16t2 + -16t2 Combine like terms: 123t + -123t = 0 615 + -123t + -16t2 = 0 + 16t2 + -16t2 615 + -123t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 615 + -123t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -38.4375 + 7.6875t + t2 = 0 Move the constant term to the right: Add '38.4375' to each side of the equation. -38.4375 + 7.6875t + 38.4375 + t2 = 0 + 38.4375 Reorder the terms: -38.4375 + 38.4375 + 7.6875t + t2 = 0 + 38.4375 Combine like terms: -38.4375 + 38.4375 = 0.0000 0.0000 + 7.6875t + t2 = 0 + 38.4375 7.6875t + t2 = 0 + 38.4375 Combine like terms: 0 + 38.4375 = 38.4375 7.6875t + t2 = 38.4375 The t term is 7.6875t. Take half its coefficient (3.84375). Square it (14.77441406) and add it to both sides. Add '14.77441406' to each side of the equation. 7.6875t + 14.77441406 + t2 = 38.4375 + 14.77441406 Reorder the terms: 14.77441406 + 7.6875t + t2 = 38.4375 + 14.77441406 Combine like terms: 38.4375 + 14.77441406 = 53.21191406 14.77441406 + 7.6875t + t2 = 53.21191406 Factor a perfect square on the left side: (t + 3.84375)(t + 3.84375) = 53.21191406 Calculate the square root of the right side: 7.294649687 Break this problem into two subproblems by setting (t + 3.84375) equal to 7.294649687 and -7.294649687.Subproblem 1
t + 3.84375 = 7.294649687 Simplifying t + 3.84375 = 7.294649687 Reorder the terms: 3.84375 + t = 7.294649687 Solving 3.84375 + t = 7.294649687 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.84375' to each side of the equation. 3.84375 + -3.84375 + t = 7.294649687 + -3.84375 Combine like terms: 3.84375 + -3.84375 = 0.00000 0.00000 + t = 7.294649687 + -3.84375 t = 7.294649687 + -3.84375 Combine like terms: 7.294649687 + -3.84375 = 3.450899687 t = 3.450899687 Simplifying t = 3.450899687Subproblem 2
t + 3.84375 = -7.294649687 Simplifying t + 3.84375 = -7.294649687 Reorder the terms: 3.84375 + t = -7.294649687 Solving 3.84375 + t = -7.294649687 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.84375' to each side of the equation. 3.84375 + -3.84375 + t = -7.294649687 + -3.84375 Combine like terms: 3.84375 + -3.84375 = 0.00000 0.00000 + t = -7.294649687 + -3.84375 t = -7.294649687 + -3.84375 Combine like terms: -7.294649687 + -3.84375 = -11.138399687 t = -11.138399687 Simplifying t = -11.138399687Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.450899687, -11.138399687}
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