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Simplifying (t2 + 4) + (5t2 + -2t + -7) = 0 Reorder the terms: (4 + t2) + (5t2 + -2t + -7) = 0 Remove parenthesis around (4 + t2) 4 + t2 + (5t2 + -2t + -7) = 0 Reorder the terms: 4 + t2 + (-7 + -2t + 5t2) = 0 Remove parenthesis around (-7 + -2t + 5t2) 4 + t2 + -7 + -2t + 5t2 = 0 Reorder the terms: 4 + -7 + -2t + t2 + 5t2 = 0 Combine like terms: 4 + -7 = -3 -3 + -2t + t2 + 5t2 = 0 Combine like terms: t2 + 5t2 = 6t2 -3 + -2t + 6t2 = 0 Solving -3 + -2t + 6t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + -0.3333333333t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.3333333333t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.3333333333t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.3333333333t + t2 = 0 + 0.5 -0.3333333333t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.3333333333t + t2 = 0.5 The t term is -0.3333333333t. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333t + 0.02777777779 + t2 = 0.5 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333t + t2 = 0.5 + 0.02777777779 Combine like terms: 0.5 + 0.02777777779 = 0.52777777779 0.02777777779 + -0.3333333333t + t2 = 0.52777777779 Factor a perfect square on the left side: (t + -0.1666666667)(t + -0.1666666667) = 0.52777777779 Calculate the square root of the right side: 0.726483157 Break this problem into two subproblems by setting (t + -0.1666666667) equal to 0.726483157 and -0.726483157.Subproblem 1
t + -0.1666666667 = 0.726483157 Simplifying t + -0.1666666667 = 0.726483157 Reorder the terms: -0.1666666667 + t = 0.726483157 Solving -0.1666666667 + t = 0.726483157 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + t = 0.726483157 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + t = 0.726483157 + 0.1666666667 t = 0.726483157 + 0.1666666667 Combine like terms: 0.726483157 + 0.1666666667 = 0.8931498237 t = 0.8931498237 Simplifying t = 0.8931498237Subproblem 2
t + -0.1666666667 = -0.726483157 Simplifying t + -0.1666666667 = -0.726483157 Reorder the terms: -0.1666666667 + t = -0.726483157 Solving -0.1666666667 + t = -0.726483157 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + t = -0.726483157 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + t = -0.726483157 + 0.1666666667 t = -0.726483157 + 0.1666666667 Combine like terms: -0.726483157 + 0.1666666667 = -0.5598164903 t = -0.5598164903 Simplifying t = -0.5598164903Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.8931498237, -0.5598164903}
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