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Simplifying 10 = 23t + -5t2 Solving 10 = 23t + -5t2 Solving for variable 't'. Reorder the terms: 10 + -23t + 5t2 = 23t + -23t + -5t2 + 5t2 Combine like terms: 23t + -23t = 0 10 + -23t + 5t2 = 0 + -5t2 + 5t2 10 + -23t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 10 + -23t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2 + -4.6t + t2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4.6t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + -4.6t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4.6t + t2 = 0 + -2 -4.6t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 -4.6t + t2 = -2 The t term is -4.6t. Take half its coefficient (-2.3). Square it (5.29) and add it to both sides. Add '5.29' to each side of the equation. -4.6t + 5.29 + t2 = -2 + 5.29 Reorder the terms: 5.29 + -4.6t + t2 = -2 + 5.29 Combine like terms: -2 + 5.29 = 3.29 5.29 + -4.6t + t2 = 3.29 Factor a perfect square on the left side: (t + -2.3)(t + -2.3) = 3.29 Calculate the square root of the right side: 1.813835715 Break this problem into two subproblems by setting (t + -2.3) equal to 1.813835715 and -1.813835715.Subproblem 1
t + -2.3 = 1.813835715 Simplifying t + -2.3 = 1.813835715 Reorder the terms: -2.3 + t = 1.813835715 Solving -2.3 + t = 1.813835715 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.3' to each side of the equation. -2.3 + 2.3 + t = 1.813835715 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + t = 1.813835715 + 2.3 t = 1.813835715 + 2.3 Combine like terms: 1.813835715 + 2.3 = 4.113835715 t = 4.113835715 Simplifying t = 4.113835715Subproblem 2
t + -2.3 = -1.813835715 Simplifying t + -2.3 = -1.813835715 Reorder the terms: -2.3 + t = -1.813835715 Solving -2.3 + t = -1.813835715 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.3' to each side of the equation. -2.3 + 2.3 + t = -1.813835715 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + t = -1.813835715 + 2.3 t = -1.813835715 + 2.3 Combine like terms: -1.813835715 + 2.3 = 0.486164285 t = 0.486164285 Simplifying t = 0.486164285Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.113835715, 0.486164285}
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