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Simplifying (t2 + -4t + -5) + (6t2 + -7t + 3) = 0 Reorder the terms: (-5 + -4t + t2) + (6t2 + -7t + 3) = 0 Remove parenthesis around (-5 + -4t + t2) -5 + -4t + t2 + (6t2 + -7t + 3) = 0 Reorder the terms: -5 + -4t + t2 + (3 + -7t + 6t2) = 0 Remove parenthesis around (3 + -7t + 6t2) -5 + -4t + t2 + 3 + -7t + 6t2 = 0 Reorder the terms: -5 + 3 + -4t + -7t + t2 + 6t2 = 0 Combine like terms: -5 + 3 = -2 -2 + -4t + -7t + t2 + 6t2 = 0 Combine like terms: -4t + -7t = -11t -2 + -11t + t2 + 6t2 = 0 Combine like terms: t2 + 6t2 = 7t2 -2 + -11t + 7t2 = 0 Solving -2 + -11t + 7t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.2857142857 + -1.571428571t + t2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + -1.571428571t + 0.2857142857 + t2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + -1.571428571t + t2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + -1.571428571t + t2 = 0 + 0.2857142857 -1.571428571t + t2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 -1.571428571t + t2 = 0.2857142857 The t term is -1.571428571t. Take half its coefficient (-0.7857142855). Square it (0.6173469384) and add it to both sides. Add '0.6173469384' to each side of the equation. -1.571428571t + 0.6173469384 + t2 = 0.2857142857 + 0.6173469384 Reorder the terms: 0.6173469384 + -1.571428571t + t2 = 0.2857142857 + 0.6173469384 Combine like terms: 0.2857142857 + 0.6173469384 = 0.9030612241 0.6173469384 + -1.571428571t + t2 = 0.9030612241 Factor a perfect square on the left side: (t + -0.7857142855)(t + -0.7857142855) = 0.9030612241 Calculate the square root of the right side: 0.950295335 Break this problem into two subproblems by setting (t + -0.7857142855) equal to 0.950295335 and -0.950295335.Subproblem 1
t + -0.7857142855 = 0.950295335 Simplifying t + -0.7857142855 = 0.950295335 Reorder the terms: -0.7857142855 + t = 0.950295335 Solving -0.7857142855 + t = 0.950295335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.7857142855' to each side of the equation. -0.7857142855 + 0.7857142855 + t = 0.950295335 + 0.7857142855 Combine like terms: -0.7857142855 + 0.7857142855 = 0.0000000000 0.0000000000 + t = 0.950295335 + 0.7857142855 t = 0.950295335 + 0.7857142855 Combine like terms: 0.950295335 + 0.7857142855 = 1.7360096205 t = 1.7360096205 Simplifying t = 1.7360096205Subproblem 2
t + -0.7857142855 = -0.950295335 Simplifying t + -0.7857142855 = -0.950295335 Reorder the terms: -0.7857142855 + t = -0.950295335 Solving -0.7857142855 + t = -0.950295335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.7857142855' to each side of the equation. -0.7857142855 + 0.7857142855 + t = -0.950295335 + 0.7857142855 Combine like terms: -0.7857142855 + 0.7857142855 = 0.0000000000 0.0000000000 + t = -0.950295335 + 0.7857142855 t = -0.950295335 + 0.7857142855 Combine like terms: -0.950295335 + 0.7857142855 = -0.1645810495 t = -0.1645810495 Simplifying t = -0.1645810495Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.7360096205, -0.1645810495}
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