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Simplifying 9t2 + 15t + 1 = 0 Reorder the terms: 1 + 15t + 9t2 = 0 Solving 1 + 15t + 9t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.1111111111 + 1.666666667t + t2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + 1.666666667t + -0.1111111111 + t2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + 1.666666667t + t2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + 1.666666667t + t2 = 0 + -0.1111111111 1.666666667t + t2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 1.666666667t + t2 = -0.1111111111 The t term is 1.666666667t. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667t + 0.6944444447 + t2 = -0.1111111111 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667t + t2 = -0.1111111111 + 0.6944444447 Combine like terms: -0.1111111111 + 0.6944444447 = 0.5833333336 0.6944444447 + 1.666666667t + t2 = 0.5833333336 Factor a perfect square on the left side: (t + 0.8333333335)(t + 0.8333333335) = 0.5833333336 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (t + 0.8333333335) equal to 0.763762616 and -0.763762616.Subproblem 1
t + 0.8333333335 = 0.763762616 Simplifying t + 0.8333333335 = 0.763762616 Reorder the terms: 0.8333333335 + t = 0.763762616 Solving 0.8333333335 + t = 0.763762616 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = 0.763762616 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = 0.763762616 + -0.8333333335 t = 0.763762616 + -0.8333333335 Combine like terms: 0.763762616 + -0.8333333335 = -0.0695707175 t = -0.0695707175 Simplifying t = -0.0695707175Subproblem 2
t + 0.8333333335 = -0.763762616 Simplifying t + 0.8333333335 = -0.763762616 Reorder the terms: 0.8333333335 + t = -0.763762616 Solving 0.8333333335 + t = -0.763762616 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = -0.763762616 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = -0.763762616 + -0.8333333335 t = -0.763762616 + -0.8333333335 Combine like terms: -0.763762616 + -0.8333333335 = -1.5970959495 t = -1.5970959495 Simplifying t = -1.5970959495Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.0695707175, -1.5970959495}
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