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Simplifying (-4t2 + 2t + -7) + 3t2 + 5t + 6 = 0 Reorder the terms: (-7 + 2t + -4t2) + 3t2 + 5t + 6 = 0 Remove parenthesis around (-7 + 2t + -4t2) -7 + 2t + -4t2 + 3t2 + 5t + 6 = 0 Reorder the terms: -7 + 6 + 2t + 5t + -4t2 + 3t2 = 0 Combine like terms: -7 + 6 = -1 -1 + 2t + 5t + -4t2 + 3t2 = 0 Combine like terms: 2t + 5t = 7t -1 + 7t + -4t2 + 3t2 = 0 Combine like terms: -4t2 + 3t2 = -1t2 -1 + 7t + -1t2 = 0 Solving -1 + 7t + -1t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -7t + t2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -7t + -1 + t2 = 0 + -1 Reorder the terms: 1 + -1 + -7t + t2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -7t + t2 = 0 + -1 -7t + t2 = 0 + -1 Combine like terms: 0 + -1 = -1 -7t + t2 = -1 The t term is -7t. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7t + 12.25 + t2 = -1 + 12.25 Reorder the terms: 12.25 + -7t + t2 = -1 + 12.25 Combine like terms: -1 + 12.25 = 11.25 12.25 + -7t + t2 = 11.25 Factor a perfect square on the left side: (t + -3.5)(t + -3.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (t + -3.5) equal to 3.354101966 and -3.354101966.Subproblem 1
t + -3.5 = 3.354101966 Simplifying t + -3.5 = 3.354101966 Reorder the terms: -3.5 + t = 3.354101966 Solving -3.5 + t = 3.354101966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = 3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = 3.354101966 + 3.5 t = 3.354101966 + 3.5 Combine like terms: 3.354101966 + 3.5 = 6.854101966 t = 6.854101966 Simplifying t = 6.854101966Subproblem 2
t + -3.5 = -3.354101966 Simplifying t + -3.5 = -3.354101966 Reorder the terms: -3.5 + t = -3.354101966 Solving -3.5 + t = -3.354101966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = -3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = -3.354101966 + 3.5 t = -3.354101966 + 3.5 Combine like terms: -3.354101966 + 3.5 = 0.145898034 t = 0.145898034 Simplifying t = 0.145898034Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.854101966, 0.145898034}
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