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Simplifying 13(t + -5) * 5t = 6(3t + 5) + -10 Reorder the terms: 13(-5 + t) * 5t = 6(3t + 5) + -10 Reorder the terms for easier multiplication: 13 * 5t(-5 + t) = 6(3t + 5) + -10 Multiply 13 * 5 65t(-5 + t) = 6(3t + 5) + -10 (-5 * 65t + t * 65t) = 6(3t + 5) + -10 (-325t + 65t2) = 6(3t + 5) + -10 Reorder the terms: -325t + 65t2 = 6(5 + 3t) + -10 -325t + 65t2 = (5 * 6 + 3t * 6) + -10 -325t + 65t2 = (30 + 18t) + -10 Reorder the terms: -325t + 65t2 = 30 + -10 + 18t Combine like terms: 30 + -10 = 20 -325t + 65t2 = 20 + 18t Solving -325t + 65t2 = 20 + 18t Solving for variable 't'. Reorder the terms: -20 + -325t + -18t + 65t2 = 20 + 18t + -20 + -18t Combine like terms: -325t + -18t = -343t -20 + -343t + 65t2 = 20 + 18t + -20 + -18t Reorder the terms: -20 + -343t + 65t2 = 20 + -20 + 18t + -18t Combine like terms: 20 + -20 = 0 -20 + -343t + 65t2 = 0 + 18t + -18t -20 + -343t + 65t2 = 18t + -18t Combine like terms: 18t + -18t = 0 -20 + -343t + 65t2 = 0 Begin completing the square. Divide all terms by 65 the coefficient of the squared term: Divide each side by '65'. -0.3076923077 + -5.276923077t + t2 = 0 Move the constant term to the right: Add '0.3076923077' to each side of the equation. -0.3076923077 + -5.276923077t + 0.3076923077 + t2 = 0 + 0.3076923077 Reorder the terms: -0.3076923077 + 0.3076923077 + -5.276923077t + t2 = 0 + 0.3076923077 Combine like terms: -0.3076923077 + 0.3076923077 = 0.0000000000 0.0000000000 + -5.276923077t + t2 = 0 + 0.3076923077 -5.276923077t + t2 = 0 + 0.3076923077 Combine like terms: 0 + 0.3076923077 = 0.3076923077 -5.276923077t + t2 = 0.3076923077 The t term is -5.276923077t. Take half its coefficient (-2.638461539). Square it (6.961479293) and add it to both sides. Add '6.961479293' to each side of the equation. -5.276923077t + 6.961479293 + t2 = 0.3076923077 + 6.961479293 Reorder the terms: 6.961479293 + -5.276923077t + t2 = 0.3076923077 + 6.961479293 Combine like terms: 0.3076923077 + 6.961479293 = 7.2691716007 6.961479293 + -5.276923077t + t2 = 7.2691716007 Factor a perfect square on the left side: (t + -2.638461539)(t + -2.638461539) = 7.2691716007 Calculate the square root of the right side: 2.69614013 Break this problem into two subproblems by setting (t + -2.638461539) equal to 2.69614013 and -2.69614013.Subproblem 1
t + -2.638461539 = 2.69614013 Simplifying t + -2.638461539 = 2.69614013 Reorder the terms: -2.638461539 + t = 2.69614013 Solving -2.638461539 + t = 2.69614013 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.638461539' to each side of the equation. -2.638461539 + 2.638461539 + t = 2.69614013 + 2.638461539 Combine like terms: -2.638461539 + 2.638461539 = 0.000000000 0.000000000 + t = 2.69614013 + 2.638461539 t = 2.69614013 + 2.638461539 Combine like terms: 2.69614013 + 2.638461539 = 5.334601669 t = 5.334601669 Simplifying t = 5.334601669Subproblem 2
t + -2.638461539 = -2.69614013 Simplifying t + -2.638461539 = -2.69614013 Reorder the terms: -2.638461539 + t = -2.69614013 Solving -2.638461539 + t = -2.69614013 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.638461539' to each side of the equation. -2.638461539 + 2.638461539 + t = -2.69614013 + 2.638461539 Combine like terms: -2.638461539 + 2.638461539 = 0.000000000 0.000000000 + t = -2.69614013 + 2.638461539 t = -2.69614013 + 2.638461539 Combine like terms: -2.69614013 + 2.638461539 = -0.057678591 t = -0.057678591 Simplifying t = -0.057678591Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.334601669, -0.057678591}
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