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Simplifying -16t2 + 42t + 12 = 0 Reorder the terms: 12 + 42t + -16t2 = 0 Solving 12 + 42t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(6 + 21t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + 21t + -8t2)' equal to zero and attempt to solve: Simplifying 6 + 21t + -8t2 = 0 Solving 6 + 21t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.75 + -2.625t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -2.625t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -2.625t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -2.625t + t2 = 0 + 0.75 -2.625t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -2.625t + t2 = 0.75 The t term is -2.625t. Take half its coefficient (-1.3125). Square it (1.72265625) and add it to both sides. Add '1.72265625' to each side of the equation. -2.625t + 1.72265625 + t2 = 0.75 + 1.72265625 Reorder the terms: 1.72265625 + -2.625t + t2 = 0.75 + 1.72265625 Combine like terms: 0.75 + 1.72265625 = 2.47265625 1.72265625 + -2.625t + t2 = 2.47265625 Factor a perfect square on the left side: (t + -1.3125)(t + -1.3125) = 2.47265625 Calculate the square root of the right side: 1.572468203 Break this problem into two subproblems by setting (t + -1.3125) equal to 1.572468203 and -1.572468203.Subproblem 1
t + -1.3125 = 1.572468203 Simplifying t + -1.3125 = 1.572468203 Reorder the terms: -1.3125 + t = 1.572468203 Solving -1.3125 + t = 1.572468203 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = 1.572468203 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = 1.572468203 + 1.3125 t = 1.572468203 + 1.3125 Combine like terms: 1.572468203 + 1.3125 = 2.884968203 t = 2.884968203 Simplifying t = 2.884968203Subproblem 2
t + -1.3125 = -1.572468203 Simplifying t + -1.3125 = -1.572468203 Reorder the terms: -1.3125 + t = -1.572468203 Solving -1.3125 + t = -1.572468203 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = -1.572468203 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = -1.572468203 + 1.3125 t = -1.572468203 + 1.3125 Combine like terms: -1.572468203 + 1.3125 = -0.259968203 t = -0.259968203 Simplifying t = -0.259968203Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.884968203, -0.259968203}Solution
t = {2.884968203, -0.259968203}
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