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Simplifying 16 = (4t + -7)(3t + -12) Reorder the terms: 16 = (-7 + 4t)(3t + -12) Reorder the terms: 16 = (-7 + 4t)(-12 + 3t) Multiply (-7 + 4t) * (-12 + 3t) 16 = (-7(-12 + 3t) + 4t * (-12 + 3t)) 16 = ((-12 * -7 + 3t * -7) + 4t * (-12 + 3t)) 16 = ((84 + -21t) + 4t * (-12 + 3t)) 16 = (84 + -21t + (-12 * 4t + 3t * 4t)) 16 = (84 + -21t + (-48t + 12t2)) Combine like terms: -21t + -48t = -69t 16 = (84 + -69t + 12t2) Solving 16 = 84 + -69t + 12t2 Solving for variable 't'. Combine like terms: 16 + -84 = -68 -68 + 69t + -12t2 = 84 + -69t + 12t2 + -84 + 69t + -12t2 Reorder the terms: -68 + 69t + -12t2 = 84 + -84 + -69t + 69t + 12t2 + -12t2 Combine like terms: 84 + -84 = 0 -68 + 69t + -12t2 = 0 + -69t + 69t + 12t2 + -12t2 -68 + 69t + -12t2 = -69t + 69t + 12t2 + -12t2 Combine like terms: -69t + 69t = 0 -68 + 69t + -12t2 = 0 + 12t2 + -12t2 -68 + 69t + -12t2 = 12t2 + -12t2 Combine like terms: 12t2 + -12t2 = 0 -68 + 69t + -12t2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. 5.666666667 + -5.75t + t2 = 0 Move the constant term to the right: Add '-5.666666667' to each side of the equation. 5.666666667 + -5.75t + -5.666666667 + t2 = 0 + -5.666666667 Reorder the terms: 5.666666667 + -5.666666667 + -5.75t + t2 = 0 + -5.666666667 Combine like terms: 5.666666667 + -5.666666667 = 0.000000000 0.000000000 + -5.75t + t2 = 0 + -5.666666667 -5.75t + t2 = 0 + -5.666666667 Combine like terms: 0 + -5.666666667 = -5.666666667 -5.75t + t2 = -5.666666667 The t term is -5.75t. Take half its coefficient (-2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. -5.75t + 8.265625 + t2 = -5.666666667 + 8.265625 Reorder the terms: 8.265625 + -5.75t + t2 = -5.666666667 + 8.265625 Combine like terms: -5.666666667 + 8.265625 = 2.598958333 8.265625 + -5.75t + t2 = 2.598958333 Factor a perfect square on the left side: (t + -2.875)(t + -2.875) = 2.598958333 Calculate the square root of the right side: 1.61212851 Break this problem into two subproblems by setting (t + -2.875) equal to 1.61212851 and -1.61212851.Subproblem 1
t + -2.875 = 1.61212851 Simplifying t + -2.875 = 1.61212851 Reorder the terms: -2.875 + t = 1.61212851 Solving -2.875 + t = 1.61212851 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + t = 1.61212851 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + t = 1.61212851 + 2.875 t = 1.61212851 + 2.875 Combine like terms: 1.61212851 + 2.875 = 4.48712851 t = 4.48712851 Simplifying t = 4.48712851Subproblem 2
t + -2.875 = -1.61212851 Simplifying t + -2.875 = -1.61212851 Reorder the terms: -2.875 + t = -1.61212851 Solving -2.875 + t = -1.61212851 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + t = -1.61212851 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + t = -1.61212851 + 2.875 t = -1.61212851 + 2.875 Combine like terms: -1.61212851 + 2.875 = 1.26287149 t = 1.26287149 Simplifying t = 1.26287149Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.48712851, 1.26287149}
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