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Simplifying -16t2 + 144t + 224 = 0 Reorder the terms: 224 + 144t + -16t2 = 0 Solving 224 + 144t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(14 + 9t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(14 + 9t + -1t2)' equal to zero and attempt to solve: Simplifying 14 + 9t + -1t2 = 0 Solving 14 + 9t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -14 + -9t + t2 = 0 Move the constant term to the right: Add '14' to each side of the equation. -14 + -9t + 14 + t2 = 0 + 14 Reorder the terms: -14 + 14 + -9t + t2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -9t + t2 = 0 + 14 -9t + t2 = 0 + 14 Combine like terms: 0 + 14 = 14 -9t + t2 = 14 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = 14 + 20.25 Reorder the terms: 20.25 + -9t + t2 = 14 + 20.25 Combine like terms: 14 + 20.25 = 34.25 20.25 + -9t + t2 = 34.25 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 34.25 Calculate the square root of the right side: 5.852349955 Break this problem into two subproblems by setting (t + -4.5) equal to 5.852349955 and -5.852349955.Subproblem 1
t + -4.5 = 5.852349955 Simplifying t + -4.5 = 5.852349955 Reorder the terms: -4.5 + t = 5.852349955 Solving -4.5 + t = 5.852349955 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 5.852349955 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 5.852349955 + 4.5 t = 5.852349955 + 4.5 Combine like terms: 5.852349955 + 4.5 = 10.352349955 t = 10.352349955 Simplifying t = 10.352349955Subproblem 2
t + -4.5 = -5.852349955 Simplifying t + -4.5 = -5.852349955 Reorder the terms: -4.5 + t = -5.852349955 Solving -4.5 + t = -5.852349955 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -5.852349955 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -5.852349955 + 4.5 t = -5.852349955 + 4.5 Combine like terms: -5.852349955 + 4.5 = -1.352349955 t = -1.352349955 Simplifying t = -1.352349955Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.352349955, -1.352349955}Solution
t = {10.352349955, -1.352349955}
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