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Simplifying -16t2 + 112t + -112 = 0 Reorder the terms: -112 + 112t + -16t2 = 0 Solving -112 + 112t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(-7 + 7t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-7 + 7t + -1t2)' equal to zero and attempt to solve: Simplifying -7 + 7t + -1t2 = 0 Solving -7 + 7t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 7 + -7t + t2 = 0 Move the constant term to the right: Add '-7' to each side of the equation. 7 + -7t + -7 + t2 = 0 + -7 Reorder the terms: 7 + -7 + -7t + t2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -7t + t2 = 0 + -7 -7t + t2 = 0 + -7 Combine like terms: 0 + -7 = -7 -7t + t2 = -7 The t term is -7t. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7t + 12.25 + t2 = -7 + 12.25 Reorder the terms: 12.25 + -7t + t2 = -7 + 12.25 Combine like terms: -7 + 12.25 = 5.25 12.25 + -7t + t2 = 5.25 Factor a perfect square on the left side: (t + -3.5)(t + -3.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (t + -3.5) equal to 2.291287847 and -2.291287847.Subproblem 1
t + -3.5 = 2.291287847 Simplifying t + -3.5 = 2.291287847 Reorder the terms: -3.5 + t = 2.291287847 Solving -3.5 + t = 2.291287847 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = 2.291287847 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = 2.291287847 + 3.5 t = 2.291287847 + 3.5 Combine like terms: 2.291287847 + 3.5 = 5.791287847 t = 5.791287847 Simplifying t = 5.791287847Subproblem 2
t + -3.5 = -2.291287847 Simplifying t + -3.5 = -2.291287847 Reorder the terms: -3.5 + t = -2.291287847 Solving -3.5 + t = -2.291287847 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = -2.291287847 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = -2.291287847 + 3.5 t = -2.291287847 + 3.5 Combine like terms: -2.291287847 + 3.5 = 1.208712153 t = 1.208712153 Simplifying t = 1.208712153Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.791287847, 1.208712153}Solution
t = {5.791287847, 1.208712153}
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