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Simplifying 12t2 + -15t + -4t = 5 Reorder the terms: -15t + -4t + 12t2 = 5 Combine like terms: -15t + -4t = -19t -19t + 12t2 = 5 Solving -19t + 12t2 = 5 Solving for variable 't'. Reorder the terms: -5 + -19t + 12t2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -19t + 12t2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + -1.583333333t + t2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + -1.583333333t + 0.4166666667 + t2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + -1.583333333t + t2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + -1.583333333t + t2 = 0 + 0.4166666667 -1.583333333t + t2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 -1.583333333t + t2 = 0.4166666667 The t term is -1.583333333t. Take half its coefficient (-0.7916666665). Square it (0.6267361108) and add it to both sides. Add '0.6267361108' to each side of the equation. -1.583333333t + 0.6267361108 + t2 = 0.4166666667 + 0.6267361108 Reorder the terms: 0.6267361108 + -1.583333333t + t2 = 0.4166666667 + 0.6267361108 Combine like terms: 0.4166666667 + 0.6267361108 = 1.0434027775 0.6267361108 + -1.583333333t + t2 = 1.0434027775 Factor a perfect square on the left side: (t + -0.7916666665)(t + -0.7916666665) = 1.0434027775 Calculate the square root of the right side: 1.021470889 Break this problem into two subproblems by setting (t + -0.7916666665) equal to 1.021470889 and -1.021470889.Subproblem 1
t + -0.7916666665 = 1.021470889 Simplifying t + -0.7916666665 = 1.021470889 Reorder the terms: -0.7916666665 + t = 1.021470889 Solving -0.7916666665 + t = 1.021470889 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.7916666665' to each side of the equation. -0.7916666665 + 0.7916666665 + t = 1.021470889 + 0.7916666665 Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000 0.0000000000 + t = 1.021470889 + 0.7916666665 t = 1.021470889 + 0.7916666665 Combine like terms: 1.021470889 + 0.7916666665 = 1.8131375555 t = 1.8131375555 Simplifying t = 1.8131375555Subproblem 2
t + -0.7916666665 = -1.021470889 Simplifying t + -0.7916666665 = -1.021470889 Reorder the terms: -0.7916666665 + t = -1.021470889 Solving -0.7916666665 + t = -1.021470889 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.7916666665' to each side of the equation. -0.7916666665 + 0.7916666665 + t = -1.021470889 + 0.7916666665 Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000 0.0000000000 + t = -1.021470889 + 0.7916666665 t = -1.021470889 + 0.7916666665 Combine like terms: -1.021470889 + 0.7916666665 = -0.2298042225 t = -0.2298042225 Simplifying t = -0.2298042225Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.8131375555, -0.2298042225}
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