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Simplifying (3t + -7)(4t + 4) + 5t + -7 = 0 Reorder the terms: (-7 + 3t)(4t + 4) + 5t + -7 = 0 Reorder the terms: (-7 + 3t)(4 + 4t) + 5t + -7 = 0 Multiply (-7 + 3t) * (4 + 4t) (-7(4 + 4t) + 3t * (4 + 4t)) + 5t + -7 = 0 ((4 * -7 + 4t * -7) + 3t * (4 + 4t)) + 5t + -7 = 0 ((-28 + -28t) + 3t * (4 + 4t)) + 5t + -7 = 0 (-28 + -28t + (4 * 3t + 4t * 3t)) + 5t + -7 = 0 (-28 + -28t + (12t + 12t2)) + 5t + -7 = 0 Combine like terms: -28t + 12t = -16t (-28 + -16t + 12t2) + 5t + -7 = 0 Reorder the terms: -28 + -7 + -16t + 5t + 12t2 = 0 Combine like terms: -28 + -7 = -35 -35 + -16t + 5t + 12t2 = 0 Combine like terms: -16t + 5t = -11t -35 + -11t + 12t2 = 0 Solving -35 + -11t + 12t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -2.916666667 + -0.9166666667t + t2 = 0 Move the constant term to the right: Add '2.916666667' to each side of the equation. -2.916666667 + -0.9166666667t + 2.916666667 + t2 = 0 + 2.916666667 Reorder the terms: -2.916666667 + 2.916666667 + -0.9166666667t + t2 = 0 + 2.916666667 Combine like terms: -2.916666667 + 2.916666667 = 0.000000000 0.000000000 + -0.9166666667t + t2 = 0 + 2.916666667 -0.9166666667t + t2 = 0 + 2.916666667 Combine like terms: 0 + 2.916666667 = 2.916666667 -0.9166666667t + t2 = 2.916666667 The t term is -0.9166666667t. Take half its coefficient (-0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. -0.9166666667t + 0.2100694445 + t2 = 2.916666667 + 0.2100694445 Reorder the terms: 0.2100694445 + -0.9166666667t + t2 = 2.916666667 + 0.2100694445 Combine like terms: 2.916666667 + 0.2100694445 = 3.1267361115 0.2100694445 + -0.9166666667t + t2 = 3.1267361115 Factor a perfect square on the left side: (t + -0.4583333334)(t + -0.4583333334) = 3.1267361115 Calculate the square root of the right side: 1.768257931 Break this problem into two subproblems by setting (t + -0.4583333334) equal to 1.768257931 and -1.768257931.Subproblem 1
t + -0.4583333334 = 1.768257931 Simplifying t + -0.4583333334 = 1.768257931 Reorder the terms: -0.4583333334 + t = 1.768257931 Solving -0.4583333334 + t = 1.768257931 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4583333334' to each side of the equation. -0.4583333334 + 0.4583333334 + t = 1.768257931 + 0.4583333334 Combine like terms: -0.4583333334 + 0.4583333334 = 0.0000000000 0.0000000000 + t = 1.768257931 + 0.4583333334 t = 1.768257931 + 0.4583333334 Combine like terms: 1.768257931 + 0.4583333334 = 2.2265912644 t = 2.2265912644 Simplifying t = 2.2265912644Subproblem 2
t + -0.4583333334 = -1.768257931 Simplifying t + -0.4583333334 = -1.768257931 Reorder the terms: -0.4583333334 + t = -1.768257931 Solving -0.4583333334 + t = -1.768257931 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4583333334' to each side of the equation. -0.4583333334 + 0.4583333334 + t = -1.768257931 + 0.4583333334 Combine like terms: -0.4583333334 + 0.4583333334 = 0.0000000000 0.0000000000 + t = -1.768257931 + 0.4583333334 t = -1.768257931 + 0.4583333334 Combine like terms: -1.768257931 + 0.4583333334 = -1.3099245976 t = -1.3099245976 Simplifying t = -1.3099245976Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.2265912644, -1.3099245976}
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