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Simplifying -5t2 + 12t + 14 = 17 Reorder the terms: 14 + 12t + -5t2 = 17 Solving 14 + 12t + -5t2 = 17 Solving for variable 't'. Reorder the terms: 14 + -17 + 12t + -5t2 = 17 + -17 Combine like terms: 14 + -17 = -3 -3 + 12t + -5t2 = 17 + -17 Combine like terms: 17 + -17 = 0 -3 + 12t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 0.6 + -2.4t + t2 = 0 Move the constant term to the right: Add '-0.6' to each side of the equation. 0.6 + -2.4t + -0.6 + t2 = 0 + -0.6 Reorder the terms: 0.6 + -0.6 + -2.4t + t2 = 0 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + -2.4t + t2 = 0 + -0.6 -2.4t + t2 = 0 + -0.6 Combine like terms: 0 + -0.6 = -0.6 -2.4t + t2 = -0.6 The t term is -2.4t. Take half its coefficient (-1.2). Square it (1.44) and add it to both sides. Add '1.44' to each side of the equation. -2.4t + 1.44 + t2 = -0.6 + 1.44 Reorder the terms: 1.44 + -2.4t + t2 = -0.6 + 1.44 Combine like terms: -0.6 + 1.44 = 0.84 1.44 + -2.4t + t2 = 0.84 Factor a perfect square on the left side: (t + -1.2)(t + -1.2) = 0.84 Calculate the square root of the right side: 0.916515139 Break this problem into two subproblems by setting (t + -1.2) equal to 0.916515139 and -0.916515139.Subproblem 1
t + -1.2 = 0.916515139 Simplifying t + -1.2 = 0.916515139 Reorder the terms: -1.2 + t = 0.916515139 Solving -1.2 + t = 0.916515139 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + t = 0.916515139 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + t = 0.916515139 + 1.2 t = 0.916515139 + 1.2 Combine like terms: 0.916515139 + 1.2 = 2.116515139 t = 2.116515139 Simplifying t = 2.116515139Subproblem 2
t + -1.2 = -0.916515139 Simplifying t + -1.2 = -0.916515139 Reorder the terms: -1.2 + t = -0.916515139 Solving -1.2 + t = -0.916515139 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + t = -0.916515139 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + t = -0.916515139 + 1.2 t = -0.916515139 + 1.2 Combine like terms: -0.916515139 + 1.2 = 0.283484861 t = 0.283484861 Simplifying t = 0.283484861Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.116515139, 0.283484861}
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