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Simplifying 5t2 + -13t + 5 = 0 Reorder the terms: 5 + -13t + 5t2 = 0 Solving 5 + -13t + 5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1 + -2.6t + t2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.6t + -1 + t2 = 0 + -1 Reorder the terms: 1 + -1 + -2.6t + t2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.6t + t2 = 0 + -1 -2.6t + t2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.6t + t2 = -1 The t term is -2.6t. Take half its coefficient (-1.3). Square it (1.69) and add it to both sides. Add '1.69' to each side of the equation. -2.6t + 1.69 + t2 = -1 + 1.69 Reorder the terms: 1.69 + -2.6t + t2 = -1 + 1.69 Combine like terms: -1 + 1.69 = 0.69 1.69 + -2.6t + t2 = 0.69 Factor a perfect square on the left side: (t + -1.3)(t + -1.3) = 0.69 Calculate the square root of the right side: 0.830662386 Break this problem into two subproblems by setting (t + -1.3) equal to 0.830662386 and -0.830662386.Subproblem 1
t + -1.3 = 0.830662386 Simplifying t + -1.3 = 0.830662386 Reorder the terms: -1.3 + t = 0.830662386 Solving -1.3 + t = 0.830662386 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + t = 0.830662386 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + t = 0.830662386 + 1.3 t = 0.830662386 + 1.3 Combine like terms: 0.830662386 + 1.3 = 2.130662386 t = 2.130662386 Simplifying t = 2.130662386Subproblem 2
t + -1.3 = -0.830662386 Simplifying t + -1.3 = -0.830662386 Reorder the terms: -1.3 + t = -0.830662386 Solving -1.3 + t = -0.830662386 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + t = -0.830662386 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + t = -0.830662386 + 1.3 t = -0.830662386 + 1.3 Combine like terms: -0.830662386 + 1.3 = 0.469337614 t = 0.469337614 Simplifying t = 0.469337614Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.130662386, 0.469337614}
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