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Simplifying -5t2 + 14t + 3 = 3(t) Reorder the terms: 3 + 14t + -5t2 = 3(t) Solving 3 + 14t + -5t2 = 3t Solving for variable 't'. Reorder the terms: 3 + 14t + -3t + -5t2 = 3t + -3t Combine like terms: 14t + -3t = 11t 3 + 11t + -5t2 = 3t + -3t Combine like terms: 3t + -3t = 0 3 + 11t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.6 + -2.2t + t2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -2.2t + 0.6 + t2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + -2.2t + t2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -2.2t + t2 = 0 + 0.6 -2.2t + t2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 -2.2t + t2 = 0.6 The t term is -2.2t. Take half its coefficient (-1.1). Square it (1.21) and add it to both sides. Add '1.21' to each side of the equation. -2.2t + 1.21 + t2 = 0.6 + 1.21 Reorder the terms: 1.21 + -2.2t + t2 = 0.6 + 1.21 Combine like terms: 0.6 + 1.21 = 1.81 1.21 + -2.2t + t2 = 1.81 Factor a perfect square on the left side: (t + -1.1)(t + -1.1) = 1.81 Calculate the square root of the right side: 1.345362405 Break this problem into two subproblems by setting (t + -1.1) equal to 1.345362405 and -1.345362405.Subproblem 1
t + -1.1 = 1.345362405 Simplifying t + -1.1 = 1.345362405 Reorder the terms: -1.1 + t = 1.345362405 Solving -1.1 + t = 1.345362405 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1' to each side of the equation. -1.1 + 1.1 + t = 1.345362405 + 1.1 Combine like terms: -1.1 + 1.1 = 0.0 0.0 + t = 1.345362405 + 1.1 t = 1.345362405 + 1.1 Combine like terms: 1.345362405 + 1.1 = 2.445362405 t = 2.445362405 Simplifying t = 2.445362405Subproblem 2
t + -1.1 = -1.345362405 Simplifying t + -1.1 = -1.345362405 Reorder the terms: -1.1 + t = -1.345362405 Solving -1.1 + t = -1.345362405 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1' to each side of the equation. -1.1 + 1.1 + t = -1.345362405 + 1.1 Combine like terms: -1.1 + 1.1 = 0.0 0.0 + t = -1.345362405 + 1.1 t = -1.345362405 + 1.1 Combine like terms: -1.345362405 + 1.1 = -0.245362405 t = -0.245362405 Simplifying t = -0.245362405Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.445362405, -0.245362405}
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